As for tacit resignation for failing to assume office, the report is that Siri did accept it and took the name Gregory XVII.  [including from the purported FBI docuмent cited by Paul Williams].  So it's not quite the same as never assuming office.  Now, once a Pope assumes office, there's no human means to legitimately remove him (except of course by killing him). 

My problem has to do with the next 30 years of his life, in which he just went back home to Genoa and conducted business as usual for the rest of his life. He never claimed to be pope or attempted to perform the duties of the papacy. This is a sign that he did not consider himself to be the pope, and Canon Law indicates this kind of behavior as a form of tacit renunciation of the office. If you read canon 188 of the code of Canon Law (1917 code), it tells us how someone may tacitly resign from office. It lists several scenarios that constitute tacit renunciation (among them is a public renunciation of the Catholic Faith, which Mr. Bergoglio should pay attention to, but I digress), and one of them (#2) is if the cleric does not assume the office within the time appointed by law, or if there is no law, the time appointed by the ordinary.

I know.  I saw this before, but this refers to "not assuming" the office in the first place.  We had a number of Popes "check out" and spend most of their time hunting and with mistresses.  Heck, if only Bergoglio had done the same, checked out and done nothing.  We'd be better off.

That depends on what you mean by "assuming" an office. I think it's more than saying you accept an office. It seems in the Code that it means "assuming the duties of the office", something Cardinal Siri never did with the papacy. Obviously, if someone says they accept an office and then never fulfill the duties of the office, that's a problem that the Code is dealing with in the canon I cited.
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We had a number of Popes "check out" and spend most of their time hunting and with mistresses.
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But they didn't tell people to call them "Your Eminence" instead of "Your Holiness", they didn't wear red instead of white, and they didn't call themselves "Cardinal X". They also didn't -- which is a far greater problem -- allow someone else to claim to be pope at the same time, unchallenged, and even accept the other person as pope. Also, they were accepted by the whole Church as pope during their reign, something not the case with Cardinal Siri.
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In any case, if Cardinal Siri were really pope for all those years, accepting the subsequent Vatican 2 "popes" as popes would be an act of schism, severing him from the Church and voiding his office.
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EDIT slight changes

Just getting to this topic now, but I certainly agree with you, Matthew. Sedevacantism (and sedeprivationism, for that matter) are seriously problematic, but I've talked with so many adherents to those opinions who believe that they're not only certain, but that it's obvious that they are. Many people build their entire worldviews on it.

Let's suppose for the sake of argument that Cardinal Siri accepted the papacy that dark night, of October 26, 1958. He became Pope Gregory XVII. Then, according to our hypothesis, he was threatened with some dire catastrophe if he didn't resign. He resigned out of fear, which would be invalid, and remained pope during the election of Roncalli.
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So far, so good. He's still pope.
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My problem has to do with the next 30 years of his life, in which he just went back home to Genoa and conducted business as usual for the rest of his life. He never claimed to be pope or attempted to perform the duties of the papacy. This is a sign that he did not consider himself to be the pope, and Canon Law indicates this kind of behavior as a form of tacit renunciation of the office. If you read canon 188 of the code of Canon Law (1917 code), it tells us how someone may tacitly resign from office. It lists several scenarios that constitute tacit renunciation (among them is a public renunciation of the Catholic Faith, which Mr. Bergoglio should pay attention to, but I digress), and one of them (#2) is if the cleric does not assume the office within the time appointed by law, or if there is no law, the time appointed by the ordinary.
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Now, we can argue that this law does not apply to the pope, but still, it's a reflection of a basic idea of reality, that being appointed to some office including the papacy is not like getting a library card that you can stick in your wallet and forget about for the next several decades, and just go home and live your life as if nothing happened, while still retaining that office. Once Cardinal Siri left Rome after the conclave and went back to Genoa and continued to live as a cardinal, he made it clear that he had no intention of claiming to be pope or performing the office of the papacy. To say that this would invalidate the elections of Montini, Luciani and Wojtyla is like saying they couldn't receive the papacy because Cardinal Siri had received the papacy many years earlier and had stuck it in his back pocket and forgotten it was there, so they couldn't receive it. It's an idea that just doesn't make a whole lot of sense.

It doesn't add up. The Siri hypothesis is stupid and solves nothing. It must have originated in the minds of 10 year olds. Not very bright 10 year olds, I might add.