This text was originally published in 2010. In May 2013, in response to multiple inquiries about crossing the border between the European Union and Ukraine in the Carpathian Mountains region by bicycle and/or on foot, it was expanded (and moved from its original location to this blog). Now it includes information on border checkpoints between Poland, Slovakia, Hungary, Romania, and Ukraine in the vicinity of the Carpathian Mountains.
Last updated: February 3, 2019 (checked for and corrected "dead links"; please bear in mind that National Border Guard Service of Ukraine appears to have a strange habit of renaming URLs on their website quite frequently, so if you stumble on a dead link to their website, try navigating it from the homepage).
The map below shows border crossing points along the EU-Ukraine border that are reasonably close (for someone riding a bicycle) to the Carpathian Mountains.