The cooling systems in the 308/Mondial are barely adequate--they are "just enough" to get the job done when they are in good working order, and unlike a modern car, they don't have much extra capacity in "reserve." So a partially clogged radiator, marginal thermostat, improperly bled system or leaking expansion tank cap, etc. can cause the car to overheat.
The most common things that seem to go wrong are:
1. Fans not coming on
Cause: usually the fuseblocks. Sometimes the thermostatic switch in the radiator. Easy to check. Just pull the two leads off the thermostatic switch at the bottom corner of the back side of the radiator. Use a connector or a wire with alligator clips to connect these two leads together. Turn on the ignition switch (car can be stone cold). BOTH fans should come on. If they don't, the issue is probably the fuseblock. Could also be a fuse, or a relay. Each fan has its own fuse and relay. The left fan is also activated by the A/C so it has a second relay. If they come on when you connect the leads, but not when the car is hot, the switch in the radiator is bad. This can be replaced with an expensive part in a yellow horsie box from Ferrari, or even better you can get a 180° switch (slightly cooler) from David Feinberg at Ferrari Service of Bedford in NH.
2. System not bled completely
Bleeding the air out of a 308 is a little bit of a pain. Not hard, just takes a while to do it right and it really is necesary to get it right. If there is too much air in the system, it expands too much when the car gets hot and forces too much coolant into the expansion tank. It overflows. Then when it cools, the car has less coolant and more air. Then the next time it is warmed up, the situation is a little worse. It just gets worse and worse. Instructions to bleed are here. I actually bleed the air out of my radiator now and then as a matter of course. When the car is hot, even if shut off, the system is pressurized. So, pull it in the garage, shut it off, pop the front lid, grab a paper towel, open the front bleeder screw and let the air out, catching the coolant drips with the towel. Now close it up. Takes about 30 seconds and you will be shocked that you can get air out of the system virtually anytime you do it. If you put a neoprene washer or rubber o-ring on the bleed screw, it can be finger tight and won't leak or break, as those fragile brass screws are known to do.
If you keep bleeding the system and continue to get a lot of air out of it every time you drive, and the car runs hot, you may be getting air into the system someplace, such as a bad head gasket. I have heard of cars that needed to have the head bolts torqued to repair this issue.
3. Bad expansion tank cap (i.e. "radiator cap")
This one is very common. A very cheap part and you wouldn't believe how often these fail. Mostly just because the gasket gets old and "smushed" so that the cap starts to leak before the system gets to pressure. If this happens, the water can boil at a lower temp. Result: the car overheats. Get a new one cheap....and it's even the right one for your car...from Ricambi here or from Ferrari Service of Bedford.
4. Clogged radiator
It happened to me a couple years ago in the Mondial when deteriorating rubber hoses clogged the radiator with small rubber particles. I had mine re-cored for about $350 and now I have essentially a brand-new radiator. More often in older cars is a radiator which is clogged due to corrosion in the system and sometimes a radiator can be repaired by a rad shop by "rodding" it. (They take the top off the radiator by unsoldering it, then run a thin metal rod down each cooling tube to scrape out the gunk.) However, some shops will not rod an old radiator like this because they are afraid the metal is too thin and fragile. Especially when it says Ferrari on the top. I chose to re-core mine as a result. Drain the cooling system, pull the hose off the top of the radiator and look in there with a flashlight. You can see the top of the tubes and get a good idea what is going on. Re-coring means that they replace the cooling section in the middle of the radiator with brand new metal, and recycle the top and bottom of your exisiting radiator with all the mounting brackets, etc. It's much cheaper than buying an OEM radiator, and the newer cores are more efficient than older ones anyway because of a better design. There is absolutely nothing wrong with re-coring a radiator as long as the shot that does it knows what they are doing.
5. Bad Thermostat
I'm listing this one last because it seems very rare that a 308 overheats from a bad thermostat. More often, they seem to not fully close so the car takes forever to warm up in cool weather. But in theory, a bad thermostat that doesn't open will not allow coolant to reach the radiator...and that is bad. If your radiator gets hot....the thermostat is probably fine. One thing to be careful about is that your car actually has a thermostat. An old trick with some cars that run hot is to remove the thermostat entirely to increase flow to the radiator. It won't work on a 308/Mondial because it uses a special kind of "dual throw" thermostat which directs coolant in one direction when closed and another direction when open. Hence, without any thermostat at all, the coolant is not directed properly to the radiator. (Some will get there, but not enough). Removing the thermostat from a 308 is a no no and definitely will not help keep it cool!!
For any car that is running hot, I would simply work my way down this list. Start with a new expansion tank cap (cheap insurance and takes 2 seconds to install), then add some coolant, bleed it correctly, check the fan operation and take it from there.