The DS18B20 devices start communicating by pulling the data line low briefly. When it releases the data line it should rise to a logic high. This is why you need a pullup resistor.
You really need an oscilloscope to determine the correct pullup resistor size.
The specification for a logic high says: "The lower of 5.5 or VDD + 0.3". As you add more devices on the bus the pullup needs to be smaller to achieve a logic high on the data line.
Look at the data line with a scope and I'll bet that it never comes up to 5V when all 20 devices are connected.
Did it work when you tested the system on the bench?
Oh, and a dumb question, but you do have a common ground with the PSU and the Arduino?
Thank you all for your amazing ideas and suggestions. After toying around with all the ideas, two of them worked for me.
Splitting the sensors on more than one Arduino Pin: this works although it means that more cabling is required and the code becomes increasingly messy as more sensors are added.
Decreasing the pull-up resistor value: this worked like a charm! I don't have the knowledge to do exact calculations but found that a 1k ohm resistor worked for 20 sensors on a single serial bus line.
I'm sure that the first solution has many added benefits to it but for the sake of simplicity, I went with the second option.
