I am authored on a manuscript currently under review from a summer internship program. The lab was super small (just one post-doc and one lab tech) and I had a significant amount of previous research experience using the same/similar techniques. Because of my experience and them needing all the help they could get I was allowed to do a ton of work which is why I think I have received authorship.
Just because you do a particular program does not mean you are going to get published. Its all going to depend on the lab, the PI, and if you get lucky. The program I did was a very large program, but I have not heard of any other students getting published.
To keep anonymity, I'm not going to mention my program, but if you want to know, PM me.
And as gettheleadout said, I do not think any program says you cannot get published.