Is it possible to repair our sagging ceiling tiles and, if so, how?

Can you tell how they are mounted? Are they attached to firing strips or solid wood like plywood. You may be able to use finish nails in between the seams and reattach them to the supports.

Make sure the sagging is not due to a leak or too much moisture in the air, this is a common problem if they are in a high humidity location. How are they supported? If they are on a grid system you will need to pull them down to access the wiring so you can shorten them to bring that section up to meet the rest of the rooms tiles. If they are glued up the glue may be failing & in need of fresh glue to reattach.

Check out what's above!

Check to see if the dropped ceiling has a finished ceiling underneath. If it does, remove the panels and the framing, patch and paint. Or try updating the ceiling and replacing the whole thing with shiplap. This is a perfect time to do some remodeling.

Are these not a fire hazard? I thought this was not the thing to use nowadays.

Sorry to be rude - do not mean to be.

They look like they are tongue and groove. Take a flashlight, with a putty knife or something flat gently pull down at the gap and look into the ceiling to see how they are attached.

we had moisture problems in our living room due to an ice dam on the eaves trough and leaking. The insurance company had the whole ceiling torn out and they were these type of tiles. Underneath was a tongue and groove solid fir ceiling. Did't like it so we had the contractor install a suspended 2x4 ceiling--love it !!!

I have the exact same kind of ceilings in several rooms of my house! I also had one that did that. If you try to straighten them, they crack or break. If you try to nail them, it pops back down, so you would need washers to make the nail heads huge. Nothing works. I had a drywall guy come in and I thought he would replace the whole ceiling, but he actually just took off the really saggy tiles and put the drywall over (under?) the rest of the tiles. It looks amazing and it cost $400.00. (California - probably less in other states.) He textured it, and left the painting to me. The other bedroom with that kind of ceiling is starting to show signs that it will be next. I'm hoping the living room tiles will stay up longer, since it's a much larger space and I like having original features in there. My house was built in 1936. Looking at your photo again, mine were more bent than yours, but the drywall is probably still a good solution for you, too.

If they are attached directly to the ceiling it sounds like the glue is failing which happens, I am glad it is not a moisture problem this is way easier to fix then. There are tubes of Liquid Nails which works nice & easy with a caulk gun if you have enough room to get the caulk in between the ceiling & tile otherwise look for some other tubes of adhesive but I don't think it is as good as the caulk. Which I have used for putting up plastic ceiling tiles in our small bathroom which would have needed a gut job so I scraped away any rough spots or peeling paint & glued them up, that was 11 years ago and still like how it looks.

Re-tack them and paint the entire ceiling all white but be sure the paint has primer built in📌🎄