Well, I guess you already know well that knowing one language doesn't automatically enable you to speak a similar language. However, I'm in the exact situation you are asking about, as I'm living in a Slovak border town. :) My vote is for Slovak, but not for technical reasons. When the Slovak and Czech countries were one, a larger part of the media was in Czech, so Slovak speakers were exposed to the language anyway. Even after splitting, Slovaks still watch Czech movies and TV series, and listen to Czech bands, as well as their own. Unfortunately, Czechs lost the Slovak exposure after the split, so even people living close to the border find it difficult to understand Slovak. Even though they are the same language group and fairly close, there are differences in basic vocabulary (compare the words for do and say/speak, as well as common farewells and months of the year) plus a couple of grammatical differences here and there. Basically, in my case I can understand enough Czech and substitute a few words but my spoken Czech is quite poor compared to my spoken Slovak. I've often mistaken a Czech word for a Slovak word simply because it's in use here. One neat side-effect of Slovak is that you'll recognise words in other Slavic languages and will be able to make a fairly educated guess at the grammar. Technically the grammar is complex and there are a lot of details to learn, but the vocabulary is surprisingly consistent. In the cases of both Czech and Slovak, you'll have to learn a handful of new sounds and change your rhythm of speaking.

Slovak :)