Yes, it can happen. 'dropping' does not mean anything, other than you get to breathe easier and pee more. "dropping' doesn't have any relation to labor or how fast or easy your labor will be. It is usually only first pregnancies that drop. Some women never drop. Some women drop early, etc.
As with most things pregnancy related, this is just one of those things that, in the end, is a non-factor.
That said, the baby can also change positions and cause the feelings you are having.
You are getting close. Good luck!