I cross that bridge (Bull's Island) from time to time on my bike. If no one else is on the bridge, it's kinda silly not to ride your bike. If there are people walking in either direction, it's rude to ride your bike. If there's lots of people, I walk the whole way. If there's only one or two walkers, I'll ride towards them and get off well in advance. The "cop" in Frenchtown used to chase down people who rode their bikes across the bridge there.

The bridge across the river at Lambertville/New Hope really needs to be walked. Way too many pedestrians. One of the other bridges (Stockton?) needs to be walked too, and you kind of have to wait behind elements of the bridge with your bike and let people pass coming from the other way. The Millford bridge is pretty good (we usually ride it), and the bridge in Riegelsville was designed by John Roebling, the designer of the Brooklyn Bridge. It's a cool little cable suspension bridge that can scare the crap out of you if you don't focus on your line. We always ride it.

If you use good judgement, crossing any of these bridges by bike is never an issue. Unfortunately, too many people just don't exercise (or even have) good judgement.