Danish studies of traffic accidents at priority intersections have shown a particular type of accidents. In these accidents a car driver supposed to give way has collided with a bicycle rider on the priority road. Often the involved car drivers have maintained that they did not see the bicycle until immediately before the collision even though the bicycle must have been clearly visible. Similar types of accidents have been the subject of studies elsewhere. In literature they are labelled "looked-but-failed-to-see", because it seems clear that in many cases the car drivers have actually been looking in the direction where the other parties were but have not seen (i.e. perceived the presence of) the other road user. This paper describes two studies approaching this problem. One study is based on 10 self-reported near accidents. It does show that "looked-but-failed-to-see" events do occur, especially for well experienced drivers. The other study based on Gap Acceptance shows that the car driver acceptance of gaps towards cyclists depends on whether or not another car is present. Hypotheses for driver perception and for accident countermeasures are discussed.