How many words do dogs understand?

On my walk yesterday I was wondering how many words (nouns, verbs) some of my dogs knew. I easily counted to 20 before I decided to think about something else. Then last night I opened up The Super Book of Useless Information, and found a story about a border collie named Chaser who has a vocabulary of 1,022 words. I’ve found that if I keep the communications with dogs simple -- maybe 1 to 3 words phrases -- they’re pretty smart.