If (allowing for Leap Years) I had 366 stoves, I could probably fit them into my tiny flat - but I'd have to move into one of my landlord's cowsheds!
I heard if you keep them in separate cabinets, they take longer to reproduce. It doesn't stop them, but it takes longer. Or maybe it takes longer for you to notice. A serious answer to the title question, is basically like any other hobby or fun time. If it's hurting anyone(including yourself), or interfering with you taking care of your responsibilities, yes there's a problem.
Well, now I've sold the 123R to @presscall, I'm down to a Coleman Feather 442, an ex-Army Burmos 2 pint paraffin, two Optimus 80s, a Primus gas cylinder stove, a Trangia spirit burner in a BCB Osprey set, and a BCB Fire Dragon stove, plus a Coleman single mantle lantern, two Tilley lamps, a Campingaz cylinder lantern, a hurricane lamp, and an Optimus 305 petrol blow-lamp. "Only four?", he says . . . sheesh . . . When all the windmills let the lights go out, this ex-Boy Scout will 'Be Prepared'!
I think that once you have all variants of all stoves and you are still buying stoves and getting fidgety if you cant find one you like on ebay it is a good time to visit the men in white coats (you know they have fuel powered medical sterilization equipment that can be hidden under an oversized coat on the way out)
If you can keep track of, or at least list, all of your stoves, you probably do not have a problem. I find stoves that I forgot about. I counted the stoves on 2 shelves a few years ago, and it was more than fifty. Just those shelves. @z1ulike even posted threads of long unopened boxes.
Is this a real question? Probably, because there seems to be some consensus that it is possible to discern a difference between someone who just happens to have a few stoves, and someone who has a pathological addiction. But these are two very wide apart extremes. Where is the tipping point? Is there a way work out precisely where the tipping point lies? Maybe a useful analogy is sand. Almost everyone can understand what is meant by a distinction between a random handful of grains of sand on the one hand, and a heap of sand on the other. But what is the precise point where adding just one more single grain of sand will turn an accumulated assemblage of sand grains into a 'heap'? If anyone can answer that then maybe they can answer @mlkimmel92's question too ... Just one more of life's great imponderables
As far as a problem goes, I think the actual amount of stoves is far less important than the other aspects I mentioned, the same way someone who only has a couple drinks a day can still have a drinking problem.
I found that it happened for me somewhere after ONE(1). It's all such a blur.... kind of like Tequila