Well, if you're really only interested in twentieth-century history then you *could* construct a degree at Cambridge that would pretty much only encompass that. As has been said, you would have to do one paper that predates 1750 in your Part I but there is no reason that the other four couldn't be from among these five twentieth-century papers:
Paper 6: British Political & Constitutional History since 1867
Paper 11: British Economic & Social History since c. 1870
Paper 18: European History since 1890
Paper 23: The West and the Third World since the First World War to the present day
Paper 24: The History of the United States from 1865
You also have to do a Themes & Sources paper for which you write a 5,000 word essay. The two papers that are going to be on offer over the next few years that relate specifically to the twentieth-century are 'Politics of Memory in Germany after 1945' (English sources) and 'World War II and its legacy in France' (French sources).
Part II (your final year) is harder to comment on as it depends entirely on who is around and running papers. Despite that, it is still probable that you could select papers than only cover the twentieth-century. At the moment, for instance, by writing a dissertation on a specific topic of interest (anything you can find someone to supervise you for), taking Paper 7 The rise of the secret world: Governments and intelligence communities since c.1900 and doing any of the following Special Subjects:
# Class, Party and the Politics of Social Identity in England, 1914-1945 (K)
# Churchill, Roosevelt, Stalin and the 'Grand Alliance', 1940-1945 (L)
# TE Lawrence and Gertrude Bell: Britain and the Arabs 1914-1922 (M)
# Martin Luther King Jr and the Civil Rights Movement (Q)
All that being said, I would really recommend branching out at university as you may just find that you grow to love another period of History. I'd recommend looking at the faculty website (
http://www.hist.cam.ac.uk) and seeing if anything there takes your fancy. All the faculty's reading lists are available to anyone and there is also plenty of information detailing the structure of the course.