The quality factor (Q) relates to the theoretical impedance bandwidth of an antenna. Such a bandwidth could only be realized using an ideal (lossless, perfectly matched) network to interface the antenna to its terminating resistance. The Q is therefore particularly useful for deriving the limiting bandwidth of fixed, compact antennas. The Q factors of short dipoles - the most fundamental compact element - are calculated from different analytical and numerical methods. The lowest Q arises from the induced EMF method where a larger feed gap leads to a larger theoretical bandwidth for thin wire elements. This method is known to be rigorous but it is a challenge to check the result because no other theoretical or numerical methods are suitable for such thin structures, and it is difficult to realize this gap feed for a physical measurement. The results for thicker dipoles, from popular numerical solvers, are included.