The situation in which you are receiving routes but are unable to use them to send traffic—a black hole—is pretty much the worst thing that can happen to a multihomed network. All other permutations of receiving routes and being able to send packets and receive return packets are better:
This is when everything works, so obviously it’s the preferred situation.
If for some reason the routes don’t make it (for instance, because the BGP session is down), but you can still send traffic, a default route would still work. Even though the problem is reason for concern, the impact is limited.
A downed line or ISP failure are serious problems, but the impact should still be limited for a multihomed network: all traffic is rerouted over another connection, so everything still works, but the available bandwidth is lower than usual.
In this situation, your router happily continues to send packets on their way towards nothingness, even if other routes that actually do work are available.
The really bad part is that multihoming doesn’t protect you from black holes: you have to intervene manually or wait until the problem clears up. The first thing you should do is perform some traceroutes using your own addresses. If you do this on a border router, specify one of the addresses the router has from your own blocks. Save this ...