[radagastly]

I'm not so certain that Bilbo did not feel the need to respond to the summons. From The Fellowship of the Ring A Long Expected Party:

"Well, I've made up my mind, anyway. I want to see mountains again, Gandalf--mountains.; and then find somewhere where I can rest. In peace and quiet, without a lot of relatives prying around, and a string of confounded visitors hanging on the bell.

And then later in the same scene:

What fun! What fun to be off again, off on the Road with dwarves! This is what I have really been longing for, for years! Good-bye!" he said looking at his old home and bowing to the door. "Good-bye, Gandalf!"

Bilbo was clearly restless when he said this. That's an unusual feeling for someone who is a hundred and eleven years old, even if they are a Hobbit. This longing for adventure had obviously been re-stirring in him for some time, at least long enough to arrange his affairs so that Frodo would inherit Bag End when he left, probably many years before that.

Later in the book, in the chapter Many Meetings, Bilbo says:

"I have thought several times of going back to Hobbiton for it; but I am getting old, and they would not let me: Gandalf and Elrond, I mean. They seemed to think that the Enemy was looking high and low for me, and woud make mincemeat of me, if he caught me tottering about in the Wild."

Gollum's wanderings in the Wild, searching for the Ring brought him, not to the Shire, but to Mordor and Sauron where he was caught and tortured for information. If Bilbo had been allowed to wander off from Rivendell, he may well have done the same.