Assuming these are cross-headed screws...
1. Check which you have, Phillips or Pozi-headed screws.
In our field of interest, we only encounter Phillips in some flat-pack furniture and plasterboard ('drywall') screws. They are designed to cam out once the screw is set, which is why they are common in manufactured goods. They are also intended to be 'use once, discard afterwards' because the camming-out inevitably damages the head.
It's a bit hard to describe, but Phillips have a more sharply defined cross shape, and the drivers are more pointy and marginally thinner than Pozi. IMHO, they are nasty things, best put straight into the metal recycling box without any attempt to actually use them, but then I used to see them a lot in electronic equipment.
Pozi, are more easily identified, because there are extra mini-splines at 45 deg. to the main cross. This is clear on the drivers, and there are usually thin lines (at 45) stamped on the head of the screw too.
2. If they are Pozidriv screws, do they have single- or double- start threads?
Double-start means two intertwined threads along the screw, rather than just one spiral. They are good for speed on-site, but they require more torque, and (as I have found out the hard way) they are unsuitable for certain tasks such as door hinges and holding down plywood floor coverings because they can unscrew themselves when subjected to "wiggling" forces (moving a door or footfalls on flooring). They work best in longer lengths for a given size - short ones are just silly, and shouldn't really be made in the first place, to avoid mistakes in use (DAMHIKT). The double-start threads work best in softwood, intended for framing or other site applications.
The ones with a self-cutting tip/thread (either/or obvious serrations or a cutting edge ground along the conical tip of the screw) usually have an even weaker grip still - they drive nicely, but at the cost of cutting a bigger hole with less thread engaged per inch. Sometimes the makers compensate by exaggerating the thread compared to the shanks. These do grip, but the chances of chewing up the screwhead in use or snapping it off are usually much greater.
3. What state are the screwdriving bit(s) in?
Wear swiftly makes them useless. I have given up using ordinary bits now, hardened or otherwise. I only use decent-quality diamond-coated bits (usually Wera, because they are easy to find). They last far longer, and most importantly don't cam out of the screw head normally, as the diamond digs into the screw, locking the two together whilst torque is applied. If you don't use them, I recommend trying some - you won't go back to the ordinary ones! I have only one Phillips bit, also diamond, but identified with marker pen, so it doesn't get muddled with the rest.
4. Is the slipping clutch of the drill in good condition?
You say, "...the chuck gives up." I'm not quite sure what you mean, but the clutch on my Bosch 10.8V drill driver is noticeably worn now after 5+ years of DIY use. I don't think there is any practical fix - it's just slowly wearing out (my SDS drill is giving up too, probably also wear).
5. Finally, do you find you need to charge the battery more often?
As they age, their ability to deliver torque as well as duration falls off. It might be time...
. . .
Personally, I'd first try diamond bits, and adopt the rule that you throw any screw that has been inserted more than twice into hardwood, or that has ever had the driver slip whilst being driven. Life is too short to be rescuing damaged screws.
Also only use single-start screws for all but rough work into softwood, and wax them with a tea-light or similar before driving. Always a pilot hole in everything, except softwood framing/floorboards/decking, too!
If the above doesn't help, probably either a new battery or drill, I'm afraid.
HTH E.
