Many consider the power of the One Ring is making its wearer invisible and extending his lifespan. True, the ring was used for those purposes, but not solely, nor did it affect everyone. So, after some consideration, I came to the conclusion that these are not the powers of the One Ring, but rather the desires of the ones wearing it. Nowhere does it say the ring is to make the wearer invisible, this is simply an assumption made by Bilbo. No one, of those who used it, expected anything else of it.
Bilbo first used the ring when he desired nothing more that to disappear, and he did. Bilbo used the ring on many occasions afterwards, but always with the same intention in mind. The ring, simply provided. Frodo, too, used the ring to disappear and once to gaze at Mordor, just before the breaking of the fellowship. A reasonable claim is that Frodo used the ring at Bree, unintentionally, and it made him disappear, but that can be attributed to the fact that Frodo expected to disappear once the ring
hit his finger.
On at least one occasion, Sam used the ring to enhance his senses of hearing and sight. Sam also used the ring, unconsciously, to appear as a great Elf-warrior and intimidate an Orc at Minas Morgul. Though Sam did not know the ring would bear such effect, and did not understand that it has, he in fact wished to become an Elf-warrior, as that notion was mentioned earlier.
This theory also explains why the ring bears no affect on Tom Bombadil. Tom was not aware of the ring's invisibility power, as the others perceived it, and it makes a lot of sense Tom didn't wish to disappear, nor did he wish for anything else.
The ring is said to posses much of Sauron's power, it's a part of him--with the ring, they are one. Sauron didn't disappear when he used the ring, but rather channeled its power to other uses, something the Hobbits were not aware to be possible. And those who were, did not want, or have, a chance to wear the Ring.
The only conflicting idea to this theory is that of Gollum's first experience with the ring. It is told he
returned to his village, wearing the ring, only to discover no one can see him. He later expected nothing else of the ring, and that it provided--invisibility.
Update: Xamot suggested that it's possible Gollum too, wished to disappear when he returned to the village. As he killed his friend little earlier, maybe he feared some punishment.
Here follows another theory, to quote Bink:
An important thing you should note is that Gandalf on several occasions says the Ring gives its bearer power according to his own measure. Which is why, in my opinion, Sauron, a Maia, using the Ring was able to bring about a Great Darkness and rule Middle Earth. While to the far lesser creatures as Sam, Frodo and Gollum it only provided invisibility, enhancing, again my own little theory, a power which they already possessed as Hobbits--being able to move about unnoticed, and to disappear quickly when they wish--just like Gandalf says.