As the chinese new years day falls on a different date each year, anyone born around this time can be affected as to which animal year they were born under..for example, if you were born on January 24th 1904 your year would be the year of the cat, as the year of the dragon did not begin until Feb 16th.
For reference, here is a list of all new years dates since 1900
chinese new years dates since 1900
1900  - jan 31   (RAT)
1901 - feb 19   (OX)
1902  - feb 08   (TIGER)
1903  - jan 29   (CAT)
1904  - feb 16   (DRAGON)
1905  - feb 04   (SNAKE)
1906  - jan 25   (HORSE)
1907  - feb 13   (GOAT)
1908  - feb 02   (MONKEY)
1909  - jan 22   (ROOSTER)
1910  - feb 10   (DOG)
1911  - jan 30  (PIG)
1912  - feb 18   (RAT)
1913  - feb 06   (OX)
1914  - jan 26   (TIGER)
1915  - feb 14   (CAT)
1916  - feb 03   (DRAGON)
1917  - jan 23   (SNAKE)
1918  - feb 11   (HORSE)
1919  - feb 01   (GOAT)
1920  - feb 20   (MONKEY)
1921  - feb 08   (ROOSTER)
1922  - jan 28   (DOG)
1923  - feb 16   (PIG)
1924  - feb 05   (RAT)
1925  - jan 24   (OX)
1926  - feb 13   (TIGER)
1927  - feb 02   (CAT)
1928  - jan 23   (DRAGON)
1929  - feb 10   (SNAKE)
1930  - jan 30   (HORSE)
1931  - feb 17   (GOAT)
1932  - feb 06   (MONKEY)
1933  - jan 26   (ROOSTER)
1934  - feb 14   (DOG)
1935  - feb 04   (PIG)
1936  - jan 24   (RAT)
1937  - feb 11   (OX)
1938  - jan 31   (TIGER)
1939  - feb 19   (CAT)
1940  - feb 08   (DRAGON)
1941  - jan 27   (SNAKE)
1942  - feb 15   (HORSE)
1943  - feb 05   (GOAT)
1944  - jan 25   (MONKEY)
1945  - feb 13   (ROOSTER)
1946  - feb 02   (DOG)
1947  - jan 22   (PIG)
1948  - feb 10   (RAT)
1949  - jan 29   (OX)
1950  - feb 17   (TIGER)

1951 - feb 06   (CAT)
1952  - jan 27   (DRAGON)
1953  - feb 14   (SNAKE)
1954  - feb 03   (HORSE)
1955  - jan 24   (GOAT)
1956  - feb 12   (MONKEY)
1957  - jan 31   (ROOSTER)
1958  - feb 18   (DOG)
1959  - feb 08   (PIG)
1960  - jan 28   (RAT)
1961  - feb 15  (OX)
1962  - feb 05   (TIGER)
1963  - jan 25   (CAT)
1964  - feb 13   (DRAGON)
1965  - feb 02   (SNAKE)
1966  - jan 21   (HORSE)
1967  - feb 09   (GOAT)
1968  - jan 30   (MONKEY)
1969  - feb 17   (ROOSTER)
1970  - feb 06   (DOG)
1971  - jan 27   (PIG)
1972  - feb 15   (RAT)
1973  - feb 03   (OX)
1974  - jan 23   (TIGER)
1975  - feb 11   (CAT)
1976  - jan 31   (DRAGON)
1977  - feb 18   (SNAKE)
1978  - feb 07   (HORSE)
1979  - jan 28   (GOAT)
1980  - feb 16   (MONKEY)
1981  - feb 05   (ROOSTER)
1982  - jan 25   (DOG)
1983  - feb 13   (PIG)
1984  - feb 02   (RAT)
1985  - feb 20   (OX)
1986  - feb 09   (TIGER)
1987  - jan 29   (CAT)
1988  - feb 17   (DRAGON)
1989  - feb 06   (SNAKE)
1990  - jan 27   (HORSE)
1991  - feb 15   (GOAT)
1992  - feb 04   (MONKEY)
1993  - jan 23   (ROOSTER)
1994  - feb 10   (DOG)
1995  - jan 31   (PIG)
1996  - feb 19   (RAT)
1997  - feb 07   (OX)
1998  - jan 28   (TIGER)
1999 - feb 16   (CAT)
2000  - feb 05   (DRAGON)