Yes. But keep the drill speed as low as possible because plastic melts and will stick to your drill bits. Use a sharp bit and try stepping up in drill size rather than going straight in with a large bit. The reason for this is plastic, depending on its thickness and type, can deform as you drill - if you go straight in with a large bit then you may end up with a slightly triangular rather than round hole. Stick to HSS bits rather than spade bits etc, take it slowly and you should be fine.
When jigsawing, make sure you can turn the speed right down. I've tried to do it quickly, and ended up having to do it all again, as the plastic melts and then joins it's self back together behind the blade.
Just go as slow as you can and check that the cut line hasn't sealed up.
Depending on what you're cutting, it might be worth wedging something in behind the saw to keep the two pieces apart after they've been cut.
Acrylic (perspex) can be very tricky to drill because it tends to "self feed" and this results in the drill screwing itself through the plastic rather than cutting a circular hole. The plastic then shoots up the drill flutes spins round on the drill and probably splits as well - been there, done that etc. The "trade" fix is to stone or grind the leading edge off the drill so that it scrapes rather than cuts, but it is then useless for other materials. I would suggest drilling by hand so that if anything goes wrong you can stop!
Brown plastic parcel tape applied to both surfaces of plastic will help strip away cut material when sawing and thus reduce the "welding" effect as the cut material overheats and solidifies behind the blade.
I recently had to fit a number plate to a car. The instructions suggested that a special plastic drilling bit should be purchased, which is presumably already modified as GCR describes. If you can find one it might be useful if you have many to do.
I just used sticky pads in the end (no help to you though).
i cut a lot of perspex with a scrollsaw, and find that brown ,or clear, parcel tape helps, also cling film! if you wrap it tight. don't force the drill through hard , or it may split when the drill breaks through,