to add to the above - any leather 'conditioner" that makes the leather supple and sof is actually breaking down the leather, that is how it makes it soft. You will hear folks extol the virtues of mik oil or blackrock or the like. I too like Connolys and the only drawback (but true of anything you add) is that it will slightly darken the leather. On the other hand, if you have any scuffs or bangs, this evens out the look.
Like most things old, less is better - less oil, less conditioner etc.
Ed