The answer is: Yes.
Most often, clipping is done on horses that are working hard in the winter. By removing the winter coat you prevent the horse from overheating when worked. You then have to replace the coat with rugs.
There are four basic clip patterns:
1. The full clip - where all of the coat is removed. This is very, very seldom done - a fully clipped horse is left with no protection at all.
2. The hunter clip. A horse that is "hunter clipped" has the legs and a saddle-shaped pattern left intact and the rest of the coat removed. The patch under the saddle (generally made by putting the horse's saddle on while clipping it) is seen as increasing the horse's comfort when ridden. The legs are left fuzzy for protection when riding through vegetation or jumping. It's called a "hunter" clip because it's particularly popular with English foxhunters.
3. The trace clip. Trace clipping is removing the fur from the belly and flanks only, to about the height where the traces fall when a horse is driven. This is by far the most popular clip. It leaves enough coat on that the horse can be comfortably turned out, but shaves the parts that sweat the most.
4. The blanket clip. A rectangle is left on over the horse's back, with the flanks, belly, chest and neck clipped.
This foal has an unusual clip - the head and neck have been shaved, but this shows "clipping lines" clearly. (Source: Montanabw via Wikimedia Commons).