If you look at what computers can do in terms of storing data, you feel sort of safe.
It's $2^{32}-1$ files (that makes 4 294 967 295) for NTFS, only 4 194 304 for FAT32 and a minuscule 65 536 for FAT16 which is however obsolete, while exFAT reads "Nearly Unlimited" [2].
Now in practice, there can be something in the way of these impressive numbers. I have collided with the limits of my computer, again, in the form of the FAT32 formatted with 4kB clusters, which limits each directory to 21844 files per path [3]. A strange number (it's also 0x5554, which doesn't make it much less mysterious). It's the number of frames I captured on this day from the Plaza Mayor, thus meeting with my first failure to record a whole day-round movie of its life.