If a condom breaks inside you, is it possible for small pieces of the condom to remain inside you?

You can find them. You can look at the condom to determine whether it tore or broke into separate pieces, and consider using your fingers to feel for pieces of the condom in the vagina. Signs that might indicate that a piece of the condom remained would be increased bad-smelling vaginal discharge, irritation or discomfort. Consider emergency contraception and make sure you see a doctor for std testing if necessary.
Potentially. If the condom is no longer intact and part of it appears to be missing, you can digitally sweep the vaginal for any remnants. It is likely that vagina secretions would wash it out of the vagina.