I know that most people bullnose a slate tile using an angle grinder, but can I cut a 4" square slate tile at a 45 degree bevel? The application is a kitchen backsplash, specifically the spot where the tile transitions to the wall, and the end of the counter.
Your question is more about your personal grinding skill. A grinder is pretty rough and you could expect it to break the slate and look pretty ragged, unless you are just awesome. Perhaps a dremel tool would be ok... which will probably take much longer than you want.
A better idea would be to use a diamond bladed (a concrete blade) circular saw, because it will probably not completely shatter the slate.
If I was beveling a 4" piece, I would maybe sandwich it between to other pieces of slate with water based glue to help hold it and prevent the edges from breaking. Then soak it in water to remove the glue.
Use a tile saw or brick saw.
Some have a feature so you can tilt the table to make angled, or bevel, cuts. Otherwise you can cut a wooden block at an angle with a circular or table saw and use that as a "jig" to hold your tile against when making bevel cuts.