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Yes and yes!
Unglazed pottery... e.g. clay flower pots or unglazed ceramic tile... can be easily drilled with a carbide "masonry" drill bit. But hold on... if your pottery has a smooth, "glazed" surface like ceramic tile, you'll have a more difficult drilling chore as the surface is extremely hard. In fact, glazing is so hard that it can actually scratch metals! Something this hard needs a special type of drill bit!
Most people's first thought would be to use the aforementioned carbide masonry bit, but they would be only half right. Carbide bits are definitely able to cut through very hard materials but most carbide bits by design are meant to cut through non-brittle materials... concrete, brick and stone. Furthermore, their low-angled cutters require quite a bit of pressure to drill effectively.
Pressure is the last thing you want to use on glazed ceramic because the glazing in this respect more resembles glass... very hard, smooth but oh-so-brittle. Ask anyone with a ceramic tile kitchen floor and they will give you an ear-bending reverie on ceramic's propensity for chipping! In fact, a recurring handyman's nightmare is drilling through ceramic tile in an old bathroom and having the tile break... with no hope of exact replacement. Wake me up and pass the Tylenol P.M. … too much pressure!
Anyway, the solution is to use a special ceramic tile or glass-cutting drill bit. These bits have a totally different shape than a carbide masonry bit with a more distinct point allowing more aggressive cutting with less pressure. Fortunately, these bits are available at virtually any hardware store.
Three additional thoughts... first, in order to keep the drill bit from slipping off the mark, put a few layers of masking tape over the ceramic before drilling. Second, water can act as a lubricant and can also help keep the drill bit a little cooler, though very thin glazes really don't need the extra lubricant. Third, if the hole you are drilling is more than a quarter inch deep, clean the chips from the hole as you go. This will keep the bit cooler and also extend it's life.