My '90 Brougham had a perfect headliner when I bought it. Now small patches are beginning to sag (like the reading lamp panels and where the visor touches), especially in wet weather. But can these small patches be fixed or is it downhill from here?
On another note, is there a type of replacement headliner that won't do this?

I don't think there's really anything you can do once it starts sagging in spots. I've known some people that have tried tacking it back up with spray glue, but that doesn't last and looks tacky. You probably need a new headliner. About the new one not doing the same, think how long it took the original to sag, 16 years isn't bad. Also, the color of the roof has a lot to do with how quickly the headliner will sag, dark colors draw more heat and will make it sag faster, what color is your Brougham?

My Brougham is White, so it stays reletively cool in warm weather. However the dark blue leather seats will really make you uncomfortable on hot days. Do you think by cracking the windows and lowering the interior temp would help?

White is the least likely to sag, my white '83 still has the original un-sagging headliner. Than again, it has a lighter saddle interior. Lowering the interior temp would help the current one last longer, yes. But it's inevitable that you'll need a new one.

Have to agree with Ben. Replacement is the only option. Be sure to do the sun visors at the same time so everything matches. Since the whole thing is held together with glue, eventually all fabric headliners will sag. Now if you were willing to go with painted cardboard... (don't laugh; AMC used this strategy in the late 60s and early 70s. Then they got bought out by Chrysler. Telling, that.)

Upside, the whole job shouldn't set you back more than $200 at the outside.

Good luck,

Craig