When does a message queue become a bottleneck, you ask?

I've been using message queues for things like sending emails and handling background jobs, you know, the asynchronous stuff. They're fantastic for separating services and dealing with sudden bursts of traffic. But I'm starting to wonder when the queue itself might become the problem. Are there any particular metrics or patterns that show this is happening? And what have you done to scale or tune message queues when they get overloaded?