Bristol confirmed their place in the play-off semi-finals after running in six tries against Bedford.
Jason Spice, Junior Fatialofa and Roy Winters all touched down to give Bristol a 24-3 lead at half-time.
Second-half efforts from Fatialofa, Adrian Jarvis and Lee Robinson secured a bonus point for Paul Hull's men.
But Bristol's defence leaked 29 points as Bedford ran in five tries of their own from Sam Walsh (2), Luke Fielden, Duncan Taylor and Myles Dorrian.
Bristol now sit on top of Pool A with 19 points, four points ahead of Bedford.
They travel to Cornish Pirates on 1 May needing just a single point to guarantee top spot and a home semi-final on 8 May.
Bristol head coach Paul Hull told BBC Radio Bristol: "We targeted the first-half to make sure we started well after what happened a couple of weeks ago and we were exactly where we wanted to be.
"It was disappointing we didn't have the same intensity and same focus on individual tackles (in the second half).
"Bedford did attack well but I don't think they deserved 29 points and we contributed to that."
Bookmark with:
What are these?