Yes you can - use a diamond drill bit (actually slate is soft enough to use an appropriately sized masonry bit but you'll have to smooth the edges of the hole afterwards) with water to lubricate.
Your main issue will be layout. Do the pipes concerned have anything attached to the top of them? If not then drill away and drop the drilled tile over the top of the pipe. Make the hole bigger than the pipe to a) allow for expansion and b) allow for the fact that you won't get the pipe dead centre in your hole. Use a pipe collar to finish off.
If the pipes do have something attached (eg radiator) and you don't want to drain /remove it then you have to find another way. If the pipe concerned falls at the grout line of two tiles just drill a half sized hole in each and fit round the pipe. Its inevitable that such will work for one end but not the other. At the other end you'll need to cut a U shape in your tile for the pipe. If its 15mm you'll most likely find that the pipe collar will cover most of it.