Is the listed height on fridges really the max with legs cranked all the way up?

Your father in law is mostly right. Published heights usually assume the leveling legs are somewhere near thier full range, and they can include hinge covers or anything sticking up on top. When I put our fridge into a tight alcove last year, the spec sheet said it was taller than my opening by about the same margin you're describing, and once it was actually in place and leveled down, there was plenty of clearance. The legs on most fridges screw in and out a good inch or more, so the listed height is realy a worst case, not what you get in practice. That said, don't just trust it blindly. The way I handled it was to call the manufacturer's support line and ask what the minimum height is with the legs fully retracted. They can usually pull that up, and it's often half an inch to an inch less than the published number. Some brands also publish install guides with the actual clearance requirements, which is worth checking before you buy. A couple things to watch while you're at it. Measure the opening at several points, front to back and top to bottom, because cabinets are rarely perfectly square and the tightest spot is what matters. Also check the depth situation, since the doors will usually stick out past the boxes and you want them to clear the face frames. And leave yourself a little breathing room for airflow at the top if the install guide calls for it, since cramming it flush to the cabinet above can hurt cooling.

If the numbers still don't work even with the legs fully up, some models in the same size class do run shorter than others, so it's worth comparing a few spec sheets rather than giving up on the alcove.

Same thing happened with our kitchen redo. The fridge spec said it wouldn't clear the cabinet above, but once it was in and we screwed the legs up as far as they'd go, we ended up with extra room to spare. One trick that helped during install was pulling the front grille off, it clips on and gives you a little slack while you're tilting the box into the opening. Also worth knowing the delivery guys will usually pull the doors off if the fit is really tight, and that gets you several more inches of wiggle room going in.

Following this because we're staring at the same problem in our kitchen, our slot is tight at the top too. One thing I keep wondering about is the floor itself. Our kitchen tile sits higher than the wood floor in the dining room, and the fridge would end up half on each. If we ever redo that floor with thicker tile or something, does that eat into the height we have up top? Feels like the kind of thing nobody thinks about until the new floor goes in.

Update: took the grille off like the second poster said and cranked the legs down, and it actually clears now. Still stuck on the gap at the top though, the spec sheet doesn't say how much breathing room it needs above. How much space did you all end up leaving?