Yes, someone will inevitably lose both. You just need to ensure that that is a rare event, and that there are alternative systems in place (i.e. that losing access to one system does not prevent people from living their lives).
And there's the rub. "What do you mean, I can't ever see my data again? Why can't you reset my password?"
We know that true security means only the user has the key. But users don't all want that responsibility.
The catch there is that IBE requires a centralized, trusted key-issuing service where you need to enroll to receive your message. If that's compromised, then game over.
http://www.argreenhouse.com/society/wcan06/wcan06s4p3.pdf
Of course, you would need to be judicious about which group of key issuers you are willing to trust, but this method will at least reduce the risk. The other nice thing about this is that even if some key issuing service is compromised, the sender can force the receiver to switch services (compare to the TLS model, where dropping a CA is basically a coordination game problem).