Hello,
They can die both way. I can't talk with experience of changing dozens of them, but even if you have a spark or even if a spark tester lights up this do not mean the spark is strong.
As you know, an ignition coil has two windings, one called primary that generate voltage with the magnet and a secondary to step up the voltage to an high value required for a spark at the plug. The last part is a trigger mechanism to release that spark at the exact right timing, more often integrated in what we call coils since about the 80's (if not you have points). More useful information here
http://www.foxvalleykart.com/timing1.html
So about dying, if you have a problem with the trigger circuit it will die immediately.
You can have a cut in primary, or it is grounded, no more spark. A short of the output of secondary coil to ground no spark again.
But if you have a problem with windings isolation, it can go slowly, starting from shorting some portions (less turns = less voltage => weak spark) or working when cold but not hot for example. You can have cut in secondary and micro-sparks the bridge the gap, so again weak spark.
You can have the HV wire to the plug that has isolation degradation, again this eat energy before reaching spark plug. Or some moisture/path somewhere in secondary coil that has the same result.
As you can see, many way to fail - even if it is a reliable part on on small engine, we see often way more problems with carbs