I think a "proper" analogue theatre has two projectors. With the films split into reels, one reel is played through one projector, and then when it's time to switch to the next reel, the projectionist cues it up on the second projector, and then switches between the two. Like a DJ mixing records, I suppose.
So if one lens is broken, they need to stop the film, feed in the second reel, then carry on playing. Hence the 20 minute break.
I may be wrong about this. Other commenters seem to be former projectionists; I'm just some guy who's seen Fight Club ;)
Many still do run changeover systems, though.
(In case you're interested, I work in an indie cinema in the UK, and we're really scared about the costs of changing to digital, which we will need to do in the next 12 months)
On Youtube, if you search for words like "theatre projector changeover" there are some great videos of projectionists showing how it's done.
Apparently there are "platter" systems that let you splice all the reels together and project the entire film through one lens with no changeovers, but my story took place at least 15 years ago, and it was not a very big or modern theatre.
I might be misunderstanding something about cinema projection, but how does a broken lens lead to 20-minute segmentation? The only things I can think of are perhaps it has some sort of cooling mechanism (I know the bulbs do emit massive quantities of heat) that needed a chance to cool off, or they were somehow playing it on a different projector which couldn't handle a full film-reel?
Certainly a more interesting experience than the few glitches I've encountered including "Yes, we know you've been complaining since the 3rd minute of playback that we had the aspect ratio wrong, but we're only going to fix it now, 70% of the way through the film"