I originally didn't want to write this as a separate answer because swbarnes2's answer is perfect. However, I did want to add this extra info.
Without a crib bumper the worst-case nightmare scenario, that is almost impossible is a broken arm or leg without a crib bumper (and I have never heard of that happening, it would be a freak accident). With a crib bumber the worst case scenario is suffocation and I hear about that all the time.
The only time a crib bumper is appropriate is when the child's head can fit through the rails of the crib because of a deformity of the child. Even then, it's better to stick with other forms of sleep like maybe a flat crib matress on the floor instead of in a crib. This example child would be a very odd exception and not the rule. To be clear if there is no deformity, and the childs head fits through the slats then it's time to go get a new crib.
For most everyone the answer is no, crib bumpers are not safe, and should be avoided. There is a very small population of children that may need a crib bumper due to circumstances, but even then other sleeping setups should be attempted first.