When does it actually make sense to shard or partition a database?

Our main PostgreSQL database is bumping up against some scaling limits. We've tried the usual stuff, like optimizing queries and adding read replicas, but those feel like temporary fixes now. Sharding or partitioning has come up, and that feels like a huge move. So, what are the clear signs it's actually time to think seriously about sharding, not just 'we're getting bigger'? When does all that extra complexity actually make sense? And what are the big things you look at when picking a sharding strategy, like using customer ID or time? Have any less common methods worked out for other folks?