Yes! You definitely can! I do it all the time ... just put the paper in with about 1/4 of an inch or so apart, enough so that you can cut in between and when it comes out, just cut. Hope that helps! I try to make my laminating paper last as long as possible so I always do multiple things in it unless it is simply an actual entire sheet of paper.
The lamination is most durable when not cut to the edge (either by laminating one page per sheet or leaving a border of lamination around smaller items) but I still cut things to the edge all the time out of necessity and have never had a problem. However, if the signs are going to be used outside, I wouldn't cut them, as water can seep through cut edges.
I made all sorts of small cards and games out of these sheets. I loaded as much in there as possible. You need to leave enough space in between each piece of whatever so that you can cut it and not break the seal. (I used to leave just a couple of mm) I made them for kindergartners and they always held up well. You can also cut the lamination before into the size you need...but that is a little trickier as you have to line it up exactly. --but if you don't have any more to laminate and have 1/2 a sheet left - I always cut it before laminating it so I didn't waste it- as it is so expensive.
I have done this, however, you need to leave ample space to make sure the sheets stay together. You many need to leave 1/8" or more depending on the thickness of the sheets you are laminating, I hope this helps.