Are New Trays Suppose to Feel Like They Don't Fit Properly?

As convenient as Invisalign is, it is frequently helpful to ease patients into Invisalign. Many doctors will wait to put on attachments and even to start the process of pushing on the teeth until the second set of aligners. You mentioned the first set was "form fitting" so this is a possibility.

As the other doctors have mentioned new aligners are supposed to feel tight when they first go in and with proper wear over 2 weeks get to the point where they no longer feel like they are moving teeth and you jump to the next set.

My advice is that if you have aligner "chewies" I'd use them to seat the aligners fully on your teeth if you think there might be a possibility that this set of aligners might not fit as well as they should. 

The new set of trays should feel tighter, but they should NOT move up or down as you bite. Perhaps you did not wear your first set as regularly and consistently as you were instructed : 22/7! Or perhaps you needed to wear the first set a little longer than the minimum 2 weeks.

Great question!  The way this system works is by programming a small amount of tooth movement into each set of aligners. So the first set took your teeth from point A to point B, while the second set will take your teeth from point B to point C.  A new set of aligners should feel tight and your teeth might even get sore after you wear them for a day.  This sensation should wear off as the teeth move closer to their point C position and the programmed tooth movement in that set of aligners becomes exhausted.

From what you're saying, it sounds like the first set of aligners did not get your teeth to point B and ready to accept the second set.  This is why the second set is not fitting tightly.  Also, the more attachments you have on your teeth, the better the aligners will "snap in".  If you don't have any attachments or just one or two, you may have loose fitting aligners.  If future aligners do not fit well, talk to your orthodontist about it.  Good luck!