Alternative Character Interpretation: One could make the argument that Billy is suffering from severe PTSD and is undergoing Flashbacks instead of actually traveling through time. The Tralfamadorians could also be stated to be a non-existing coping mechanism to help deal with the trauma and the flashbacks.
Unlike most "it's all in the main character's head" interpretations, this one bears a lot of weight. After all, the Tralfamadorians look like the aliens from one of Kilgore Trout's stories, and Billy's abduction uses convoluted time-travel logic.
Additionally, the idea of being abducted by aliens and placed in a zoo was in one of Kilgore Trouts books. Relevant passage: "He got a few paragraphs into it, and then he realized that he had read it before — years ago, in the veterans' hospital. It was about an Earthling man and woman who were kidnapped by extra-terrestrials. They were put on display in a zoo on a planet called Zircon-212."
More evidence: Billy views a porn movie (or something like that... I forget the exact technology that was used) of Montana Wildhack, and that she had apparently disappeared at some point. This could have expanded into her being abducted alongside him and not coming back (in his mind).
Even more: After first being captured in the war, Billy sees one American soldier being knocked around by a German soldier. The American asks, "Why me?" and the German replies, "Why you? Why anyone?". This same exchange is repeated when Billy is first kidnapped by the Tralfamadorians, with Billy as the American and a Tralfamadorian as the German.
In addition there's the fact that Billy only starts talking about his time travel after he suffers a severe head injury. The timing of all this together really is quite suggestive.
Most importantly, let's not forget that while the novel always presents Billy's time (and space) travel in a very matter-of-fact way, as if it really were happening, it is also stated near the very opening of the story that the book is really just reporting what Billy himself says happened to him — the entire book may be told from the perspective of a hidden Unreliable Narrator.
Big Lipped Alligator Moment: While watching a war film on television, Billy becomes slightly Unstuck in Time, and sees the whole thing in reverse. Before this scene, there was no indication that this was a possible effect, and nothing like it ever happens to him again.
Continuity Lockout: Averted. Despite the cameos of some characters from his earlier novels, Vonnegut gives enough context for them that you don't need to have read those books to understand this one.
Tear Jerker: The story is filled to the brim with the dead, dying, and a series of awful events that happen to one person. The comforting power of the "so it goes" mantra can only take you so far (if it takes you anywhere at all). The story is very sad.