Nope, they never stopped of their own accord and yes, it did cause a breakdown.
I was being plagued by seemingly continuous gearbox failure warnings and it was becoming silly. I told the dealer how to fix it (yes, really....) by changing the switch under the brake pedal, as this is a common cause. That cured it for a few months and then it returned. Then the selespeed computer was changed and that cured it for another few months.
Then, I was driving down the A38, in a contraflow lane. Crawling along slightly uphill, got all the bells and whistles as yet again, gearbox failure warning comes on. Left me with no forward or backwards gears and stranded, immobile. Two people pushed me into a coned-off zone (not a nice place to be with HGVs passing by you) and I phoned the AA. Whilst waiting for them, I tried the little-known 'selespeed reset procedure' which, believe it or no, involves opening and closing the driver's door 9 times in a row.
Did that procedure twice and it came back to life, albeit badly changing gears - slipping clutch at high revs. This got better, seemingly as it re-learnt its settings.
I took it back to the dealer and had a major showdown with the dealer principal as I'd absolutely had enough. Long story follows, but the upshot is that I'm trading it in for a new 54 plate one, as I have no confidence in this one, long term. If it would have been financially viable (and it's not, due to bank loan and effective negative equity on the value of my car), I'd have thrown them the keys and walked away from Fiat totally. As it isn't viable, they've pulled some strings and I'm getting a new Stilo Abarth at a *very* cheap price as I'm now effectively classed as 'staff' at the dealers....
I once seriously gave thought to taking a sign to put up in the service bay with my car's registration number on, so it could have a permanent, dedicated service bay as it seems to spend so much time there. I also took them a cake on the 21st visit to fix yet another fault.