Are easily broken slate tiles an indication of poor quality?
Slate floors are beautiful...rich colors and a great natural look. However, they need to be laid perfectly and maintained constantly. 

Make sure each tile is "back-buttered" when you install. That is to say, you can't just press them into a layer of mortar when you install them. Make sure you apply a layer of mortar to the back side of each tile before you press them into the bed of mortar...and don't get too far ahead of yourself. Make sure the mortar is pliable. Otherwise the tiles could break if they are not fully embedded. 

Immediately after the grouting, make sure you use a two step sealer process. The first seals the small holes to keep the dirt out and the second seals the surface. You'll need several coats, but slate is rough enough to keep the floor from being slippery. 

Lastly, re-seal at least once per year, depending on traffic. If the sealer wears off and dirt gets into the slate, it will show wear marks, traffic patterns, etc. and it's nearly impossible to fix.

I like slate, but lay it right and be prepared for regular maintenance.