The ring was only as powerful as its owner

[Mister Underhill]

The power of the Ring seems to rely on a combination of the relative power of its wearer and the bearer's will to power, or at least, will to domination. Galadriel's brief comments to Frodo seem most revealing:

`I would ask one thing before we go,’ said Frodo, `a thing which I often meant to ask Gandalf in Rivendell. I am permitted to wear the One Ring: why cannot I see all the others and know the thoughts of those that wear them?’

`You have not tried,' she said. `Only thrice have you set the Ring upon your finger since you knew what you possessed. Do not try! It would destroy you. Did not Gandalf tell you that the rings give power according to the measure of each possessor? Before you could use that power you would need to become far stronger, and to train your will to the domination of others. Yet even so, as Ring-bearer and as one that has borne it on finger and seen that which is hidden, your sight is grown keener. You have perceived my thought more clearly than many that are accounted wise. You saw the Eye of him that holds the Seven and the Nine. And did you not see and recognize the ring upon my finger?’

Interestingly, Galadriel implies that knowledge of what the Ring is and what it does is an important part of the equation. ‘Only thrice have you set the Ring upon your finger since you knew what you possessed,’ she says, indicating that she is aware of each of the three occasions (presumably via Nenya). Perhaps Bilbo’s ignorance of what he possessed helped shield him from the Dark Lord’s notice while he was under his nose.