Gandalf in the Lord of the Rings chapter A Long-expected Party confirms that magic rings in the general case are not too common:
Magic rings are – well, magical; and they are rare and curious.
Despite that, the Rings of Power (even those given to Men and Dwarves) are not the only magic rings in Middle-earth, as the Lord of the Rings chapter Shadow of the Past confirms:
In Eregion long ago many Elven-rings were made, magic rings as you call them, and they were, of course, of various kinds: some more potent and some less. The lesser rings were only essays in the craft before it was full-grown, and to the Elven-smiths they were but trifles – yet still to my mind dangerous for mortals. But the Great Rings, the Rings of Power, they were perilous.
Moreover, when the actual physical characteristics of the Rings are described in the Council of Elrond we see the following:
"The Nine, the Seven, and the Three," he said, "had each their proper gem. Not so the One. It was round and unadorned, as it were one of the lesser rings; but its maker set marks upon it that the skilled, maybe, could still see and read."
Because Bilbo's ring was also "round and unadorned" Gandalf can easily surmise that it was not one of the Nine, the Seven or the Three (of these only one of the Seven would have otherwise been a realistic possibility). That leaves two possibilities: it is either one of the other lesser rings, or it is the One. Of those, the balance of probability is that it is one of the lesser rings - despite being rare enough, there are still more of them than there are of the One, and the One was presumed lost.
'There I was at fault,' he said. 'I was lulled by the words of Saruman the Wise; but I should have sought for the truth sooner, and our peril would now be less.'
'We were all at fault,' said Elrond, 'and but for your vigilance the Darkness, maybe, would already be upon us. But say on!'
'From the first my heart misgave me, against all reason that I knew,' said Gandalf, 'and I desired to know how this thing came to Gollum, and how long he had possessed it. So I set a watch for him, guessing that he would ere long come forth from his darkness to seek for his treasure. He came, but he escaped and was not found. And then alas! I let the matter rest, watching and waiting only, as we have too often done.
'Time passed with many cares, until my doubts were awakened again to sudden fear. Whence
came the hobbit's ring? What, if my fear was true, should be done with it? Those things I must decide. But I spoke yet of my dread to none, knowing the peril of an untimely whisper, if it went astray. In all the long wars with the Dark Tower treason has ever been our greatest foe.'
(Council of Elrond)