If you mean top hats and such, sure, but the Panama hat, cowboy hat etc absolutely still have their plan in men's clothing, particularly if you live in the kind of climate where shade from the sun is a necessity.

I think what the parent means is a hat that's an indispensible article of formal dress. They certainly still have their place, but it's not like there's an expectation to see everyone in cowboy hats at a Texas wedding.

The only place I'm aware of where hat use as a matter of dress and appearance is still strictly adhered to is in the US military.

I think one can go farther than that... wearing a cowboy hat or a Panama hat is a bold statement. At one point it would have been a fairly unremarkable stylistic choice, but no longer.