The most correct answer is NO. Ballistic vests are not designed for dealing with knives and ice picks. Stab vests protect against these threats, but not bullets.
Of course, a nice thick vest would help protect you against slashes, just as wearing a few layers of denim jacket. Here, the ballistic vest simply provides a thick layer of extra clothing. A high tech Second Chance ballistic vest that is very thin would be poorer at this task than an older and low tech very thick vest.
I do recall that some companies make combination vests, but they are about twice as thick since they do two jobs.