I have an old house we are partially rewiring. To meet the NEC we have installed arc fault breakers where required. In one bed room the circuit breaker will not remain set. I have checked the boxes and verified all of the connections and can find no problem. Could we have a shared neutral in some of the old wiring?? Would this cause the arc fault to trip?

Or you could be having an Arc Fault?
Either way, try to troubleshoot this circuit... disconnect the circuit somewhere near the middle and see if the breaker will reset. If not, continue dividing the circuit until it clears... now you will have a much better idea of where the problem is...
Good luck...

Or you could have a neutral touching a ground somewhere.

Why did you install the arc-faults? You do not need them on old wiring to meet NEC code. These requirements are not retroactive.

Have you swapped breakers with one known to be good? Defective arc fault breakers are far from unheard of.