And yes, the obvious answer is 3/16" ... that doesn't work.
I'm installing a towel rod and hook in a bathroom that is fully tiled. The instructions with the rod & hook both said to drill a 3/16" hole for the included anchors, so I bought a Rigid 3/16" diamond tile drill bit. Drilling the first two holes went great, until I tried to fit the anchor in. Didn't fit.
Generally with drywall, the anchor is a little bigger than the hole, and you just force it in and it works fine. There's no give with the tile though.
For reference, I grabbed a caliper and started measuring things. 3/16" is 0.1875" The holes I drilled with the 3/16" bit are actually 0.225" and the widest part of the anchor is 0.245" -- almost 1/4"!
As a secondary question -- assuming I get get the right size bit -- am I going to run into trouble trying to make these holes larger? If so, is there a better way you can think of to mount the hook with the two 0.225" holes I already have?