It may just be flight school rumours, but I've also heard a similar thing to Mawright. Sources say that after four EASA theory fails it becomes extremely hard, if not impossible to find a job.
As for which airlines specify what minimums requirements, some say Ryanair and EasyJet only accept applicants with an EASA theory average of 85% or above, but then some sources reckon Ryanair will allow up to two re-takes. Supposedly Emirates and BA will only accept those with first time passes in everything, but I doubt the number of people who go straight from training to those two is very high. Flybe seem to be quite well-known to only accept those with an EASA theory average of 90% and above.
Then again, I was told a few weeks ago that some people on a well-known MPL course got right to the end of their training, and were dismissed by the airline for failing the final interview and theory assessment, so it goes to show that you really can't take your foot off the pedal when it comes to flight school performance.