When should you shard a relational database to make it bigger?

I'm working on a project and our main PostgreSQL database is starting to slow down. We've done a lot, like making queries faster and adding indexes. We even tried bigger servers a few times, but I think we'll need a distributed setup pretty soon. Sharding looks like a good option, but I'm trying to figure out when it's actually worth the trouble. When does sharding a database really help, instead of just making things harder to manage? And what are the usual problems people hit when they try to do it?