There were failures. For a list of some problems that occurred, see Appendix C at https://web.archive.org/web/20010412195518/http://www.iy2kcc.org:80/Appendices.htm (the International Y2K Cooperation Center glitch list).

Most problems were minor. 15 nuclear power plants had issues, all minor (7 in the USA, 2 or 3 in Japan, 2 in Spain, at least 1 in Russia). Since plants were being monitored for problems, the problems could be fixed, or the affected system shut down and the back-up system used instead, or the plant shut down. Banks and other financial institutions knew about the possibility, and Y2K problems were generally quickly identified as Y2K problems and fixed.

One problem that wasn't caught by the kind of attention paid to Y2K by the banking industry was the British NHS sending incorrect Down's syndrome test results to 154 pregnant women, with 2 terminations resulting (and 4 babies with Down syndrome being born to women who had been sent incorrect negative test results). It shouldn't have happened because unusual test results is one of the things that should have been watched for.

More details in my reply from a year ago in https://www.reddit.com/r/AskHistorians/comments/abcfl9/why_did_y2k_kill_my_digital_watch_but_nothing_else/