It's not particularly common for the ROM chips on a C64 to fail, but it does happen occasionally. The kernal has the highest running temperature of the ROMs, and also tends to be the most likely to fail, followed by the Character ROM. Generally the ROM chips run hotter than the CIAs but cooler than the SID, VIC and PLA.
Since cartridges bypass the default ROMs, they are useful for diagnosing ROM faults. Ultimax type cartridges are best for this purpose, as they also bypass the kernal. Kernal ROM faults seem to predominantly cause a simple black screen, thus if you are getting a black screen but an Ultimax cartridge works, a bad kernal is one possibility. Character ROM faults often cause corrupt startup graphics, though colors and borders will remain normal. If only the default character set graphics are incorrect but everything else works fine, suspect the character ROM. A Basic ROM fault most commonly causes a blank blue screen with normal borders on startup.