Dead circuits with breakers across from each other
If all the breakers on every other row go out, you lost  one of your phases.  That's not as bad as losing a neutral, in which case each 120V leg can go as high as 240V, blowing stuff up and starting fires.  All that to say, if it proves to be a wire connection problem, check your neutral too!

If both breakers in a single row go out, you may have a burned up bus bar there. 

For two rows, and not others, to go out, very unusual.  Maybe if they're the bottom two on the panel and somehow the bus bar burned up or took damage, but they usually don't make bus bars thinner near the bottom, so there's no reason for that to happen.