If I'm understanding the situation correctly: the system stopped processing updates yesterday afternoon and it wasn't a big deal, then they tried restarting it this morning to fix it and it didn't come back up, so now older NOTAMs from before the issue started are not available. Is that right?
Do airlines/planes keep a cached copy of these? How large is the NOTAM database?
Not really. And for most commercial pilots this may come down to whether or not ForeFlight caches theirs or not. I'd actually be really curious to see unofficial commentary from ForeFlight developers about this because it's a pretty unique scenario in the age of electronic flight bags.
The problem isn't keeping a copy, it's sending updates that are very time critical (for example restricted areas around POTUS, any impactful event, etc.)
Do airlines/planes keep a cached copy of these? How large is the NOTAM database?