The Digital Reader, an expert on all things ebook, posted this useful Q&A for people who own Borders ereaders, Borders tablets, or Borders ebooks, about what happens to their property now that Borders is being liquidated.
The short answer: Kobo is a separate company, and it lives on so you and your books should be okay.
However, this debacle does raise questions about the risk of “owning” digital files on a proprietary device. A paper book sits on your shelf forever. Might your digital library be at risk if your vendor goes bust?