There have been those players yes, but not with big success at the same time or not for any long period in any case. It's like only for a few years some player will excel in both, you can't make a career of being good in both, especially in modern physical game and schedule.
Evidence to the contrary... examples of players who excel at both (bolded players are current):
: reached #1 in doubles in 2010, 13 GS titles + 3 Olympic gold medals
: as above
: reached career rankings in singles and doubles simultaneously (#6 singles, #1 doubles)
Hingis: finished in doubles top 10 for 5 consecutive years (1996-2000), 9 GS titles including doing the Grand Slam in 1998
Navratilova: do I really need to give details on Martina?
Mandlikova: 27 career singles titles (incl. 4 GS), 19 career doubles titles (incl. 1 GS)
Graf: In doubles: 8 GS SF, 3 GS finals, 1 GS title, finished top 20 1986-1989.
: Finished year in top 20 - singles 9 times, doubles 7 times. Singles titles: 13. Doubles titles: 23
: reached singles career high (#12) while at doubles career high ranking (#1). 19 WTA titles in doubles, 8 in singles
: highest year-end ranking in singles and doubles occurred in same year (2012, #14 singles, #7 doubles). 12 WTA titles in doubles, 6 in singles
Zvereva: reached 19 WTA or GS singles finals between 1986 and 1999 (won 4), won 80 doubles titles between 1988 and 2002 (including 18 GS titles)
Sanchez Vicario: finished year in top 10: singles 11 times, doubles 10 times. GS titles: 4 singles, 8 doubles. WTA finals: 77 singles, 111 doubles
Sukova: 4 time GS singles finalist, 9 time GS doubles winner. Finished year end top 10: singles 1984-1989, doubles 1986-1993, 1995-1997
Novotna: finished year in top 10: singles 7 times, doubles 10 times
Shriver: finished year in top 10: singles 6 times, doubles 7 times
Davenport: 55 singles titles, 38 doubles titles. finished year in top 10: singles 10 times, doubles 7 times
: reached career highs in both at the same time (#11 in doubles, #30 in singles)
Others who have done well in both: Azarenka
, Suarez, Evert, Gigi Fernandez, Mary Pierce, Conchita Martinez, Mary Joe Fernandez, Kim Clijsters, Ai Sugiyama and so on and so forth.