You can play in just a T-Shirt if you want, but those little balls can sting like a bitch...
Nothing wrong with playing in a t shirt
However a cautionary note, I had one of these a few weeks back:
http://en.m.wikipedia.org/wiki/Olecranon_bursitis
WARNING - Link leads to explicit elbow swelling!
My elbow suddenly developed exactly what's show in the picture, and was badly infected
If your going to crawl, dive or bump into things then wear some form of elbow and/knee protection
In general wear what is comfortable for you.
Paintball jerseys are (has described) designed for paintball
They may be thinly or heavily padded, have breathability and wicking etc
I don't do speedball, but did have a light paintball jersey for a few years and occasionally word it
I play scenario and vary what I wear depending on mood & conditions
I wear combat trousers, and jeans a couple of times, t-shirts, ubacs (army wicking top with a lighter body area as it's worn with an assault vest / body armour), jackets or hoodies
For elbow protection I wear either nothing, 'sleeves' or specific paintball arm/elbow pads
For knee protection I don't normally wear anything, on occasion have worn army style knee pads - I'm not keen on these and they are only good for knealing, not for crawling or diving. A pair of my trousers has classic sniper knee pads - a large rubber panel sewn into the knee. These are good for knealing on and ideal for crawling