Depends on the "generic" e-mail address and where it has been used.
I opened a HotMail account, setup a typical profile and sent one e-mail to myself to check it. Never sent another e-mail or ever gave that e-mail address to anyone. Within a week I was getting over 100 SPAM per day. It got up to over 200 per day.
So I setup a different one, and kept the profile to a minimum and signed of for NO options and activated the SPAM filters. Never sent an e-mail using that e-mail address or told any one about it. After a few months about six SPAM messages per day were showing up. They had found it somehow and were getting through the SPAM filters.
All it takes is opening one e-mail from a friend who has forwarded you a "Joke of the Day" and your e-mail address is then on a list that gets sold around to SPAM'ers.
Another trick use is to setup list of common names and add to the list the first part of every valid e-mail address found. Then use this list to SPAM different e-mail domains. Any that are not returned as undeliverable are assumed to be valid address. So if you use your favorite "ID" with different ISP's, they will find you on each of them.
I to have noticed a sudden jump in SPAM to my public e-mail address when first starting to use this forum. If you put your e-mail address in the header so people can contact you, then any one can click on that link and get your e-mail address, including the SPAM'ers. I get over 250 per day that I have to sort through. I am sure I lose a few e-mail from people I don't recognize.