I am picky about fabric, so here are my musings. None of this is based on good data, just my observations.
First, yes, some cotton is better. And it's not just the fibers - it's also the dye, the washing/processing, and the other treatments. The processing means some cotton fabrics shrink more than others (or shrink more evenly - my H&M tshirts all got diagonal over time!), and (the hidden thing) there is a huge range of treatments applied to the cotton that still can be called "100% cotton", as in no-iron shirts that are treated with formaldehyde.
But second, high end designers are often optimizing something other than fabric quality or longevity. The companies that actually talk about their fabric quality the way you mean are Lands End and LL Bean and such (at least back in the day - unsure if that's still true) and even then, I can't tell what's hype and what's legit. Higher end designers might be trying to get the perfect drape, using thinner cotton that doesn't hold up as well, for instance.
That said, there's a fairly linear increase in quality with price at the low-medium end. I know this is silk not cotton, but my Equipment silk shirts are so much sturdier than my BR silk shirts. But I wouldn't necessarily expect a Prada silk shirt to be sturdier than the Equipment shirts.
And synthetic content makes a huge difference but it's hard to predict. I basically hate synthetics (just a quirk, not usually a value judgement). Anything that's cotton/poly blend feels cheap and plasticky to me. On the other hand, I was pleasantly surprised by my amazing Theory button-down that has a hefty amount of nylon (maybe 70% cotton?) and maintains it shape all day.
So, to conclude my rambling: some brands are definitely better fabric than others. But it's not directly related to price. Instead, I suggest feeling the fabric or collecting opinions on brands from people here. I know there are Anne Fontaine fans here. I've heard good things about Brooks Brothers shirts too. Theory is always reliable. And, I have a cotton shirt from Everlane (an online store with simple basics sourced directly from the mills and factories) that lives up to its hype too - quite good for the price.