I hit a bicycle but had the right of way. Am I still at fault?
Typically, no. If you had the right of way, you're likely not at fault for causing the accident. Even in traffic accidents between drivers of four-wheeled vehicles and bicyclists, the issue of who was liable often turns on whether a traffic law or "rule of the road" was violated by one of the parties.
In most instances, a bicycle is considered a vehicle, so bicyclists must follow the same rules of the road as automobile drivers -- they enjoy the same rights as other drivers, but they're also saddled with the same obligations. That includes yielding the right of way to other vehicles when the situation dictates doing so.
It may seem odd, given the disparity in size between a car and a bicycle -- not to mention the physical vulnerability of the bicyclist if an accident were to occur -- but a bicycle is treated just like any other vehicle in most scenarios on the road.
So, after an accident between a car and a bicycle, if an insurance claim or personal injury lawsuit comes down to the issue of fault, the success of your case may depend on who had the right of way at the time of the crash.
One way to answer this question is to figure out who was (and who wasn't) obeying all relevant traffic laws at the time of the accident. For example, let's say a traffic accident occurred in an urban setting. A bicyclist failed to stop at a red light right before getting struck by a car entering the intersection on a green light. That's a clear case where the bicyclist's blatant violation of a traffic law -- running the red light -- caused the accident. The car driver in this scenario clearly had the right of way, and would not be at fault for the accident.
It's also worth noting that, while bicyclists must obey traffic laws in most situations, there are also some traffic laws that apply only to bicyclists. Failure to follow these special traffic laws may go a long way toward proving that a bicyclist wasn't acting with reasonable care. Some examples of these laws are the requirement that bicyclists ride as near to the right side of the road as practicable, prohibitions on the carrying of packages and other passengers on a bicycle, and mandatory use of proper visibility aids when riding at night.