What happens if you obtain a cursed item while already attuned to your maximum number of attuned items (three)?

Reference: daze413’s answer to “As a monk, can I use Deflect Missiles to protect my ally?”

Scenario: Monk already has his three attuned magic items. Monk picks up a shield of missile attraction, which is a cursed item, and uses it for awhile. This should cause him to pick up the curse. Monk discards the shield of missile attraction.

If I read correctly, most magic items do not function unless attuned, and using a cursed item causes the attunement to pick up automatically (otherwise what’s the point)?

But now we’re trying to exceed the attunement limit by using a cursed item. Oops. Contradiction. Now what?