The annoying answer is "it depends." The main drivers are reliability (ie: how much risk of failure are you willing to accept) and mission life (ionizing dose is cumulative, so a 2 year vs. 10 year mission will have different requirements).

I would say you certainly need to start seriously considering at least some radiation hardening at around 600 km, but missions that can accept a large amount of risk to keep costs down still operate at that altitude with non-hardened parts. Likewise, missions with critical reliability requirements like the International Space Station use radiation hardening even down at 400 km.

The "hard" limit is probably around 1000 km, which is where the inner Van Allen Belt starts. At this altitude, hardware that isn't specifically radiation hardened will fail quickly.

The inner Van Allen Belt also has a bulge that goes down as low as 200 km (the South Atlantic Anomaly), so missions in low inclined orbits that spend a lot of time there or missions that need good reliability when flying through the SAA may also need radiation hardening at comparatively low altitudes.