Sorta. But you did not say what kind of spam you are referring to; e-mail, webdexing, forum, or otherwise. I am assuming all of them.
Yes the odds go up, but depending upon the host. It may not be much of a risk at all. For example, some hosts actively block e-mail spam even when sent directly. It all depends upon the neighborhood. For example, you can have a dedicated IP address and someone on your address block sends spam. Many blacklists will list all the IP addresses on that block especially if it is a subscriber block.
The most important thing when choosing a host is the reputation of that host. It is okay to have many sites on a single server. Under normal circumstances, this will be quite fine. But with a sleazy host, the odds of your having trouble will go up dramatically.
I used to be a web host for many years starting in the early days. I had no problems with my site owners because I screened my customers. I cared about what happened on my network and servers. I put in protections that ensured that one customer could not ruin the day of another or even ruin my reputation. I had special blocks that prevented bad behaviors which never happened.
Some hosts take the steps to make sure that their networks and servers are protected and that any customer that goes rogue are stopped immediately and shutdown. Some host don't care what you do. They just want to get paid.
Chose a good host with a good reputation.