12v 16 amps to 12v 3 amps?
It is indeed possible to supply a steady 12 V at 3 amps from a 12 V 16 amp SMPS.

The thing to watch out for with ye olde PC PSUs is that some of them need some load on other rails to be stable.

I thought, and could well be wrong, that it was the case that a load (e.g. some 12 V fan that is to hand) was required on the 12 V rail for the PSU to stay on, but it could be that you need a small load on the 5 V and/or 3.3 V rails.

Drawing a few hundred milliamps from the other rails at the same time should enable the PSU to stay on.