Isolate the lathe first then put a length of stout bar in the chuck ,put a dial indicator somewhere on the chuck and push and pull the bar ,if the indicator moves more than a tenth or two I would say you have a problem ,It may be that they can be adjusted ,it depends what you are making as to whether it really is a problem or something you can live with.
On edit
the vibration could easily be something else ,I have a lathe that shakes a bit at higher speeds and I assume that to be and out of balance chuck but as I don't really need to use the higher speeds much it is not an issue.
Also it would be a good idea to tell us what lathe this is (unless it is a Chinese POS) as someone will likely have experience of your make and model of machine.