can arduino control 1000 leds

There are a few important questions you need to ask yourself.

Do you need individual control of the LED's ? or are they turned on/off in groups ?

Is on/off enough or do you need to dimm them to have light levels in between on and off ?

How important is timing ? when daisy chaining som of the LED control chips you can run into propagation delays after a certain number of chips in the chain.

Will the Arduino boards 16 KB of program memory and 1KB of RAM be enough for your project ?

These issues might or might not be problematic, but you should consider them carefully.