Sidekiq falling over is a big one. See: <https://docs.joinmastodon.org/admin/scaling/> and <https://nora.codes/post/scaling-mastodon-in-the-face-of-an-e...>
(I have to email my own admin every few months to ckeck if things are OK.)
And during the October Revolution as hoardes arrived from birdland, things got ssssslllloooowwww globally.
Worked out eventually, but it took a bit.
Individual instances also tend to run into scaling issues, with Jerry Bell's Infosec.Exchange coming to mind. (Mostly because Jerry's discussed this a bit.) And of course individual instances can be shut down or fail in various ways. I've migrated several times myself.
I will say that most of the time things seem fine, and it's exceptionally rare for there to be truly Fediverse-wide issues.
(I've been on Mastodon / the Fediverse since 2017, for the most part quite actively.)