The rule of thumb with fluorescents was the way most went to try out replacement.
3 main components being the starter, the tube and the choke.
So, first try a new starter is the cheapest, easiest option, then if still no then try a new tube, then if that doesnt work it is probably the choke, in that case its often easier and cheaper to replace the whole fitting which often came supplied with a new tube.
Of course we have moved on from that and nowadays LED battens are all the rage and fluoros are more expensive if you can get them.
PS the tube ends blackening was often a sign of a bad tube past its life, although they could often carry on a good while after that in many instances.