When I first started making name tags, I would print the artwork, coat it with plastic, and then cut the name tag to size.
This process makes an inexpense, nice looking name tag which consists of a layer of plastic, the artwork, and a second layer of plastic. However, because cutting the name tag after the plastic has been applied does expose the edge of the paper, there are two potential problems (I say potential because out of the thousands and thousands of name tags I've made, only one of my customers has ever had any problems with the name tags).
First, if you drop the name tag into an aquarium, go swimming with it on, or expose it to a lot of moisture, the water can "wick" in through the edges that have been cut. This picture shows the back of one of my name tags:

Because I coated the artwork with plastic before cutting the name tag to size, the artwork paper is exposed along the cut edges:

To eliminate any potential problems, I've started calling my original style of name tag a "single cut" name tag (because I cut the artwork only once), and am now offering a second style of name tag which I call "double cut" (because end up cutting the artwork twice).
I cut the artwork to size, coat it with plastic, and then cut the name tag - leaving a slight edge of plastic all the way around the name tag. I thought it would be a great idea to show you a picture of a double cut name tag, but it is awfully doggone hard to get a picture where the plastic edge shows up, since it is only about 1/16" of an inch wide and is clear. The edge just kind of disappears and becomes very hard to see or photograph.
The best picture I got of the double cut edge was when I mounted the name tag on a backing board, turned on a spotlight, and zoomed in real close. This picture shows a highly magnified image of the 1/16" wide clear border around the double cut name tag. Well... actually, that is not quite accurate - what you see in this photo is the shadow of the edge, but it should give you a better idea of what I'm talking about.

When you order name tags, you can pick either the single cut or the double cut style. The double cut name tags are 25 cents more than the single cut versions. The double cut name tags are also more water resistant and more resistant to folks picking on the edges with a thumbnail.
If you have any questions, please let me know.
Rob