Can the police give someone a ticket after they let you go?

Yes. They normally only do this in a situation where more investigation is required, such as in the case of an accident, but they can mail you your ticket at a later date.

Added: Most (all?) states have a statute of limitations that governs when you must be given notice (the ticket) that you are being charged with an offense. Normally, this is anywhere from 1-3 years.