When it comes to an antenna there is one optimal set of, well, variables, if you will. Unfortunately none of these variables exist in a vacuum, any change to one will change one or more others.
For example, any number of shortened 1/4 wavelength antennas. If you use the most common method of shortening antennas, namely a loading coil, you will find the bandwidth narrow some. Further, where the loading coil is on the antenna also affects how much the antenna is shortened and efficiency as well. The further up the antenna the coil, the more efficient it is and the less of a shortening effect it has. That being said the top of the antenna isn't always the best place to put such a coil, depending on your needs...
The properties of the coil also have effects. If it is a lossy (or inefficient) coil you will see more bandwidth than a less lossy coil will have on the same antenna. The antenna itself doesn't actually have a wider bandwidth, it just appears to to your radio.
All this in conjunction to a cap hat when it comes to shortening. A capacity hat will shorten the antenna and increase the bandwidth at the same time. Further, while any shortening method decreases efficiency, it tends to be more efficient than using a loading coil for similar amounts of shortening.
All this and I'm not even touching on the half of it.
In the end the question is how much of a compromise is acceptable to you, where that compromise needs to be, and how much is needed to meet your needs...
The DB