as has been said, normal swelling and pain from a missed shot usually only lasts a day. you'll know by that point whether or not it's getting better on its own or whether it has developed into an infection or something similarly serious. as long as it's not red or hot, it's probably not infected. if it does become red or hot though, you should really get it checked out.
early in my IV career I missed a shot in my wrist (which is not a good place to shoot generally but it was the only one I could find because at the time my technique was bad, and I had really messed my veins up). after only half a day it was worse, not better. my arm hurt way worse than a regular missed shot, which is kind of subjective--but what's not subjective is that my whole wrist was red and hot, and the pain started moving up my arm. 24 hours in, the pain was all the way at my bicep, and the redness had spread halfway up my arm. clearly something was wrong so I went to the ER, afraid I would lose my hand or something! they diagnosed it as cellulitis (which is a skin infection) and lymphangitis, which means the bacteria had travelled from the wound into my lymph nodes. pretty scary shit. I had to take antibiotics and stay in the hospital for a day and they also said I was at risk for permanent joint damage due to the location of the infection (thankfully that didn't happen).
the moral of the story is: wrists can be tricky due to how skinny the veins are (not to mention all the nerves and arteries around there), always use sterile equipment and filter, and if you have a physical complication that is getting worse, go to the hospital! it may be awkward and you may even get some dirty looks but it's worth it to avoid potentially permanent complications.
