New B&Bs are opening all the time, and hostels are on the rise. There’s no network of hostels in Slovakia, though a few are affiliated to HI (hihostels.com). Bratislava has plenty of good private hostels, and in the High Tatras you can find chalet-style boarding houses (chata) scattered over the mountains, with basic dorm beds from €8 per bed. There are plenty of campsites in Slovakia, which usually rent out basic wooden huts (chata), fun if you share with friends.
In summer the cheapest accommodation is university halls; ask at the local tourist office for details. Private rooms are also cheap, and can be organized through the local tourist office as well. If you’re a student, get hold of an ISIC (International Student Identity Card), as you’ll need it to get student rates. Check websites for online booking discounts. Smaller places are often understaffed so pre-book and give an estimated arrival time, or you might find yourself locked out.