In the movie "The Princess Bride," Wesley informs Buttercup that her marriage to Humperdink never happened since she never said "I do." Thus, there would be no problem marrying him (Wesley) since she was not actually married to another man.
A similar situation was presented to Rav Moshe Feinstein in 1977. A man had come to America from Russia and had been married before his emigration. However, it was common for weddings in Russia to be done without chuppah and kiddushin, and he had separated from his so-called-wife before he moved. Could he go ahead and marry another woman without having to worry about divorcing his previous wife?
Rav Moshe ruled that at a basic level he could marry another woman, since his original marriage was likely done without chuppah and kiddushin, and even for those who worry about civil marriage having some halachic import, even the civil marriage was merely an act of registering one's name with the local authorities (which could be undone by just as simple an act). Even further, it was common for people to be separated from their spouses by the government for one reason or another, usually for relocation purposes, and thus the entire enterprise of marriage was nearly meaningless.
However, while for marriage purposes for the man Rav Moshe had no major issues, he was concerned about the first wife resurfacing, claiming that their marriage was done with all proper halachic ceremonies, and being put into a possible situation of being an aguna. Thus, Rav Moshe suggested that the man draw up a get, or at least testify before witnesses that should the woman appear he will give her a get immediately without any complaints.