I'd just try with one powerbank first and see how that goes. It'll obviously not last as long as two would (provided they'd work), but you'd at least get to figure out if it works at all.
Provided the bank can actually deliver the current you're asking from it. What is its maximum current rating - is that spec provided in the first place?
I haven't looked for the current rating, mainly because with the sketch only running the leds, the backpack last for more then 4 hours with all the lights flashing all the time. My wife and friends have used the backpack already several times, but always just with the sketch solely with the ledsprogramming.
It is so that I haven't had in mind the extra need for power from the MP3 player.
But what really puzzles me, is that the sketch as it is, commenting out all lines of the sound, compile it, upload it with the circuit setup as in the image and 2 powerbanks ... ... works like a clock....
It's when I decomment again the sound parts, compile the sketch (without problems), upload it again, the Arduino seems to protect itself after about 10-15 seconds by shutting off. Even the Voltmeter, which is running directly from the powerbank, and is in no way conected to he Arduino, goes dead.
Once I toggle the switch to shut the power down and toggle it on again, it does the same thing, shuts off, voltmeter dead, and on and on ....
I even switched out an Arduino (I bought several), and it does exactly the same.
What is a huge mystery is that it worked before, when I had it powered over the computer. I was so proud I have it on video.
It's starting to frustrate me, as I believe it has to do with power, more then it has to do with the sketch, but than again I know the sketch is far from ok (although it compiles without any error). 
Well, I agree with you that it looks more and more like a hardware issue related exclusively to the MP3 player. It's kind of a long shot, but have you tried:
- Having the MP3 player connected to the +5V and GND, buit not to the Arduino
- Having the MP3 player connected normally, but with no speakers
A slightly less long shot, but costs a few bucks: have you tried a different MP3 player?
Thanks for the tips, I'll give them a try. Just ordered another mp3 
The fact that your volt meter does not measure anything suggests that the powerbank is shut off. Maybe an emergency shut down because the current draw is too high?
Parallel speakers have a resistance of only 4 ohms. At 5 V that can give quite large current. Maybe you can put speakers in series instead?
I have just learned something new, thanks. I will give it a try
I really apreciate all your help
Update on this project:
- I've separated both speakers and only conected one (I didn't try the speakers in serie after that): no improvement
- I've tried the mp3 player wihout having speakers connected: no improvement
- I changed the mp3 player for a new one: no improvement
- I connected the +5V and GND to Arduino (as before it was separate): no improvement
- I powered the system with only 1 powerbank: it takes a lot longer (around 2 minutes) but the arduino and the voltmeter shut off again as well.
Just to be sure, I changed the sketch and used the one with only the lightprogramming ... : it worked ....loaded again the one with the sound programming ... no luck.
Tomorrow one of our friends will bring her backpack, which has the "original" cabling, exactly as it was before I made any changes to it. I'll connect the extra mp3 player I have now and load the sketch again with the sound programming in it. That's how it worked (or at least I hope it does it again). When it does, I'll try to make a list of all the changes in hardware (if any) I have done to it. If it's not that, and it hasn't have anything to do with the sketch and the way I wrote/modified it ... then I'm afraid they will only have lights on it.
The best I can do than, is try to make a little sketch to operate with a pushbutton, the DIY smokemachine I made with a mini aquarium airpump and a vapertip .. ...
That's a great idea; it'll help you eliminate the issue.
That's odd; I wouldn't expect the MP3 player to single-handedly bring a powerbank to its knees.
Same type, or also a different kind of product?
Yes, the same type 
Can't remember if we've gone through this yet, but have you verified the MP3 player works OK if it's the only module connected to an Arduino and running off a single battery pack?
Nope, haven't done this yet. Will do this today, and conect our friend's backpack to a mp3 player as well. I have an Adafruit soundmodule lying around, I'll give it a shot with that too.
Hi again, today I'm almost sure it's has something to do with the sketch.
I have just tried one of our friend's backpacks, hooked it all up, the way I did the first time (when it worked) and the result is the same as before, not working.
When I comment out everything that has to do with sound instructions/functions in the sketch it works again like a clock... with power from the computer or the powerbank.
When I power it all up, I can hear a short buzzing in the speaker, as if it gets it's power.
I can only conclude it has to do with programming it all, for some to me unknown reason.
Any further suggestions how I can go on from here?
Troubleshoot line by line, basically. First start including the libraries necessary for sound, then initialize the required objects, and keep re-including lines you have commented out until you hit the problem.
Ok, thanks. I'll let you know the progress
Hi again, I had to leave this proyect on standby a bit, due to work.
It's very clear to me now that it's a matter of power, mAh...
I hooked up the mp3 module to the arduno board, with only a small sound sketch that I got from the DFPlayer library, on the computer through its USB port, and worked like a charm.
As soon as I hooked it up on my powerbank, nothing happens.
Finally I got a datasheet of the powerbanks and they run only 2600mAh, with 1A, 5V.
So far I have found out that:
The arduino itself already uses around 500mAh.
The standby function of the mp3 player uses 20mAH (I have no idea what it uses to play sound, haven't found that data yet).
Over the computer's USB, the voltage meter shows 5,02V - 4,98V and it works
Over the powerbank, the voltage meter shows 4,87V, the led on he MP3 player doesn't light up (it should when the sound starts to play) and nothing happens, not even a change in the voltage meter.
The speaker is a 5V, 3W, 8Ohm little speaker (no idea what the mAh usage is) but AND I MAY BE WRONG, if the speaker is 3W, with 5V, the amperage should be 600mAh (5V x 600mAh = 3W).
So :
500mAh for the arduino + 20mAh for the MP3 + 600mAh for the speaker = 1120 mAh, which is too much for the powerbank to handle if not fully loaded (I think).
The USB Port of the computer will always give a steady V, mAh or W.
Am I right?
Completely wrong?
Does anyone have a sketch that does sound and lights at the "same" time or even one after another?
Now I am really at the end of trying to find out what the heck happens ... 
Hope someone has an answer
First of all: let's get the units right. Current is measured in mA. Battery capacity is measured in mAh (number of milliamps it can deliver continuously during some number of hours).
The use of your Arduino seems high. What is connected to the board?
The total current is larger than 1000 mA. So the power bank might get damaged, might have a current limiter, or might go into emergency shutdown, or it might deliver this current outside it's specs.
The computer USB will not deliver an infinite amount of current. It will also be limited but also damage could occur at very high currents. Maybe you should not use your computer USB port for experiments (as it is pretty difficult to replace it). You would better use a USB phone charger. But not all chargers may deliver currents higher than 1000 mA. So have a look at the specs.
So my start would be to find a charger or another power supply that can deliver 2000 mA and see how that goes.
Ok, got that
I need to use the computer's USB to uplaod the sketch to the board, although only once, as I swith to the powerbank after that, but I can't passby that step.
With this latest testing, only the mp3 player and the 3W speaker are conected to the Arduino Uno.
On previous testing, also 42 leds (3812's) were connected, that flash on and off on a 60% brightness, although not all at once.
Ok, seems a good idea. The quest for a 2A phonecharger has begun.
It is probably also best to have separate powersupplies, when I hook up the leds as well. They work great with my current powerbank of 1A.
Maybe I can try to install a step-up booster that delivers 2A und still use my actual powerbanks?
Your power bank is rated for 1A 5V. You cannot boost that to 2A 5V. Maybe there is a way to get your 2 powerbanks to cooperate instead of interfere. Hope someone else can help you with that.
Could I put a step-up, after the output of my powerbank?
You could, but to what avail. ?
A step-up will increase the voltage, not the power ( current * Voltage ) . It will actually lose some power. Your powerbank probably has a step-up in it already, to get to 5v from the 4.2v - 3.7v that the liPo provides.