Can swallowed fingermails, hair, or skin get caught in your appendix?
Skin is safe to chew, and digestible. Fingernails are not digestible, so really shouldn't be swallowed if avoidable. In the medical literature, there are at least 225 cases of foreign bodies in the appendix. They include:


a metal drill bit that was ingested unintentionally 3 months earlier
pins (81  cases)
lead shot (81 cases) 
seeds (34 cases)
bones (16 cases) 
eggshell
glass
teeth
nails (hair causes problems with bezoar formation)
a die (dice)
the clinical end of a thermometer. 


Most (71%) patients were symptomatic, typically with intermittent abdominal pain for months or years. Pins were most likely to elicit symptoms (93% of cases).

So it seems there is some truth to this nugget of folk wisdom.

The above were found over many, many years. Since most cases are asymptomatic, it's likely there are many cases that never come to light, but not remarkably so (lots of appendices are removed and examined every year!). 

Foreign Bodies in the Appendix