That did happen once, not too long after 9/11 our local Air Force base was closed to anybody who didnt work there, and security was really tight. (Before this, we used to go over and watch the F-15s practise landing and takeoffs.)
But on one side of the base, there was a company installing a huge water filtration plant. We were out biking in the Crane Wildlife Reservation which borders the base, and the construction people had bulldozed a new dirt road onto the base, so we took it. And ended up on the base, and yes, we were quickly kicked off. 2 old gray haired people we are on mt. bikes, but I can't blame them for ousting us.
We also have a PAVE_PAWS radar installation here,
http://www.fas.org/spp/military/program/track/pavepaws.htm and security is incredibly tight there. To view that incredible place, we had to apply weeks ahead of time with our SS #s and everything, and then when we got clearance, we got to tour it.
Its a fascinating place, and when the grandsons are old enough to understand it, we will arrange for them to come on another tour with us.
Jane