First, just to clear up a misunderstanding: almost all of Jumble runs locally on your device. So there’s no real difference between using jumble.social and running it locally.
And actually, this happens because Jumble uses the outbox model. It fetches data from the outbox relays of the people you follow, which means it has to connect to a lot of relays. The more people you follow, the more relays it needs. I only follow a bit over 200 people, but Jumble still connects to dozens of relays just to load my following feed.
At that point, the browser starts blocking new connections. For example, if I try to like a note, it may need to connect to a few additional relays, but those connections just sit in pending, and the like can fail.