I basically hold to Bergström's view that there has never really been a case in the history of science were multiple theories really were completely empirically equivalent.
I think the Galileo affair was one such example. At the time, there were two alternative cosmologies -- the heliocentric, advocated by Galileo perhaps too passionately; and the geocentric, particularly the model proposed decades earlier by Tycho Brahe. Each was offered as an explanation of the movement of the celestial objects. Galileo insisted that the heliocentric model was representative of the truth, but he did not have an empirical proof. This eventually lead to his legal troubles with the Church.
A more modern example would be the different interpretations of Quantum Mechanics, or at some point there were several competing versions of the string theory. The string theory itself has a competitor in the loop quantum gravity. Or think of different models of consciousness. Basically, science assumes1 that the Reality has to work, objectively, in some specific way (a.k.a. the objective/scientific truth). But until we have a definite proof one way or the other, we could have several competing theories/models of how the Reality might work.
Further update: According to the Britannica article on underdetermination,
Alternatively, one can interpret it as proposing that all the criteria of rationality and scientific method permit some means of protecting the favoured hypothesis from the apparently refuting results.
This is a fascinating proposition. Indeed, this is exactly what Galileo was doing -- defending, at the risk to his own life, the heliocentric cosmology against the apparently refuting evidence.2 This goes to the heart of the age-old question: What is truth? And if a scientific theory cannot be, generally speaking, falsified by the evidence, then what is the point of doing science?
As I understand it, the assumption behind science that our one and only objective Reality can be understood as a machine, as a very complex clockwork. To that end, scientists try to discover a model of how that machine of the Reality works under the hood -- a model of the processes driving the observable patterns (patterns like the apparent movements of the Sun, the Moon, of the stars and planets). Quine's point could be that those models don't need to be perfect, neither accounting nor being consistent with every available piece of evidence (with every available observation). Rather, we choose a model that offers what we feel is the best explanation of the available evidence.
Sometimes we might have multiple equally (un)convincing theories -- and then choosing the winner becomes as much a matter of personal taste,3 as of the amount of the evidence one is able to take into consideration. To Galileo, the discovery of moons orbiting Jupiter, as well as the mountains on the Moon, was a clear indication that there is no substantial difference between Earth and the celestial bodies -- that Earth, therefore, is but another planet orbiting the Sun. This was his truth, and as someone who dedicated his life to the pursuit of the truth, he didn't -- he couldn't -- back down. He could not un-know the truth he has uncovered, something he came to know for a fact.
1 Or, rather, this is the assumption on which the science is based upon.
2 For example, due to the primitive optics if his day, the stars appeared as disks when looking through a telescope, suggesting that they cannot be much farther from us than the planets. If that was true, then the movement of earth around the Sun would create a parallax -- an apparent movement of some stars against the backdrop of others. Yet, no star parallax could be observed.
3 The British physicist Paul Dirac, for example, claimed that it is more important that a theory be beautiful than that it conform with experiment, while Einstein stated that "the only physical theories that we are willing to accept are the beautiful ones." (link)
