Could you get sick from too many spider bites?

Depends: Most spider bite are like other insect bites and may hurt at first but afterward, like with any break in the skin, infection needs to be watched for. So cleaning is best. Some are dangerous like the well known widow and recluse bites in the usa, but there are other less known species that can cause problems like the recluse too. Generally, watch the bite and if concerned, see a physician.

Yes.: Yes depending upon the spider. A single bit by a black widow or brown recluse is going to cause some misery. One of them is too many. However, most spider bites are not anywhere near as severe and most spiders do not bite repeatedly. It is possible to have an allergic reaction to a spider bite and this too could make you "sick.".