Do Bits Have Mass?

I often joke that putting all the bits on a hard disk on one site will make it wobble because the platter would be become unbalanced.

Well, apparently the bits on a flash drive actually do have a tiny amount of mass... but not a hard drive.

I read this on the Internet, it must be true:

https://www.quora.com/Does-the-computer-become-any-bit-heavier-after-installing-a-file-on-it