Novice can last between ~10 and ~60 matches. 60 for people who don't specialise in a class and play all three about equally (while also probably not getting tutorial achievements), 10 (or even less) from people who do the tutorial achievements and then get a few achievements during their first few matches.
It's definitely possible to get engineer to lvl5 without playing a single match, may be possible to get it even higher.
I personally think that novice should be tied to match count rather than level since, especially at lower levels, it can be very different for how quickly different people level up. Having it tied to matches (50-100 at least in my opinion) would solve that, as well as not be as hard on those who specialise in a single class.
Comparing the levels in the new system to the old ones doesn't work too well either, it's so much faster even when taking into account the tripled level max.
The XP gains from levels 1 to 7 (gap between each level) are
2000, 3000, 3000, 4000, 5000, 6000, 7000
Won matches give a base of ~1000 XP with perhaps up to 500 extra for really doing your job 'well' (rebuild all the things for engineers etc). Take into account that the achievements at that level also give a lot of XP (around 2000-4000 I believe) and are fairly easy to get then you can start to see how quickly players can be kicked out of novice.
On the other hand this would make more of a difference if novice matches were actually formed by Matchmaking instead of mixing the lobbies, but that's a discussion that has been held many times elsewhere.