I've used fountain pens for many years and have seen two kinds of converter failures.
One kind is where the seal between the converter and the section fails. The traditional design is for the converter to merely slip over a nipple at the bottom of the section, and failure can occur if the plastic opening in the converter experiences wear at this point. The recommendation to not remove the converter from the pen is partly intended to minimize this problem (the other reason is that forcing ink through the section when filling the pen helps keep ink flowing). However, I suspect that this problem is something that eventually will happen to every slip on converter - its just a matter of how long before the plastic gives out. That said, I suspect that a good quality converter should last many years. I have a Cross Pen with a slip on converter that is about 30 years old that this point, and still working well.
The obvious symptom of this failure is that the converter won't stay attached to the section and/or leaks around that junction.
Incidentally, Mont Blanc uses a proprietary converter that actually screws onto the end of the section to avoid this problem. Mont Blanc pens are very expensive. Even so, I don't think the incremental cost of making the converter screw on justifies the higher price.
I have also seen the barrel of a converter crack so that ink would leak through the side. My sense is that this kind of failure is rare. Most inexpensive converters are made of some kind of semi-flexible plastic (polyethelene?), but the converter that I saw fail in this fashion was made of a harder plastic.