Potentially, both. There was an unfortunate incident today at the hospital. The patient fell from his bed. It was, of course, an avoidable accident. [ from/off - not both ]
This can really go either way. What happened is an incident. However, if it wasn't intentional, then it would also be an accident. Also, you should either say a hospital patient fell off his bed, or fell from his bed. The two adverbs should not be placed together in this instance.