I went in without issues, they just close the door at certain times and then reopen it to prevent people from overflowing the area. You have probably been given one of the times wje they'll reopen in any case.
Only been once, but I was allowed in 15 mins earlier with out any issues. I guess it depends on how busy and who is on duty.
I've been there on three separate occasions, and each time they generally started on time.
When I had my reservation during busy season I was allowed to go in whenever I wanted to so you should have nothing to worry about for your timed reservations. The museum was busy but not so busy as to cause delays in viewing of the Space Shuttle Endeavor
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