There are two factors in play here:
1) Your turn timer
2) The expel timer
#1 is simple. When your turn starts, the game credits you additional time to take your turn. The exact amount is determined algorithmically and scales based on the table's speed settings. Even if you ended your last turn in the negative, you'll be credited new time when your next turn begins. (I think they even bring your time back up to zero before adding in the new time, too.) There's a maximum amount that you can bank between turns, but it's usually quite a bit, allowing you some flexibility to have an occasional long turn. (BTW, if a game is not crediting you the additional time each turn, that's a bug with the particular game. Developers have a command they need to run at the start of each turn for BGA to give you the time, although the developers have no control over how much is given.)
If you ever run out of turn time (#1 again) completely, you'll start going into the negative. All other players at the table will have the option to try to expel you. That's where #2 comes into play.
The first time someone tried to expel you, you have a lot of leeway to still complete your turn. At least 20 seconds. However, the next time you go negative and they try to expel you, you have less (15 seconds, I think). If this happens often enough in the same
turn (edit: not game, turn), you can finally be expelled instantly, but you'll have exceeded your turn time
several times by then.
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The system is really forgiving towards players who exceed turn times occasionally. If you're exceeding it consistently, you should consider creating/joining tables with slower time settings, both to avoid being expelled and out of respect to your fellow players, who expect everyone to fulfill the same speed requirements they committed to.
Edit: The expel time resets between turns. VERY forgiving.
