Ringwraiths

[Salocin]

If the ring wraiths can feel the ring, how come that one didn't just grab Frodo when he was hiding behind that tree?

That only happened in the movie. In the books anytime the Nazgul are anywhere close to the Ring they go after it. When one of them is after Frodo in the woods in the Shire, he is "scared" off by the travelling elves singing hymes to Elbereth.

They didn't go after Bilbo because they had no idea he existed. As soon as they got the words "baggins" and "shire" out of gollum they went off to find him, but by then both reffered to Frodo.

Their sensing of the ring obviously does not have a very long range when the ring is not being used. For example, Frodo and Sam where able to sneak deep into the heart of Mordor without being noticed and even when Sam was using the ring in Cirith Ungol no Nazgul noticed, though at that time they were far away assulting Minas Tirith.