- Lives in a home in the Hollywood Hills (beneath the "D" of the Hollywood sign).
- Former roommate of Lauren Graham.
- Has an adopted son named Eyob Britton (born January 25, 2011, adopted in November, 2011) from Ethiopia.
- Considered for the role of Olivia Pope in Scandal (2012) when it planned a white woman for a leading role. Kerry Washington was ultimately cast, making her the first black woman to topline a drama in 37 years and later nominated for a Golden Globe for her performance.
- Plays the wife of the main coach character in both the movie Friday Night Lights (2004)and its TV adaptation, Friday Night Lights (2006), even though her characters have different names in the two.
- While at Dartmouth she met John Britton at Alpha Delta house, and they later married.
- Friends with Sheryl Crow.
- Nominated for the 2019 Golden Globe Award in the Best Performance by an Actress in a Limited Series or a Motion Picture Made for Television category for her role as Debra Newell in Dirty John (2018), but lost to Patricia Arquette for Escape at Dannemora (2018).
- In a February 2013 New York Times Magazine profile, Connie said that she nearly got the role of "Dorothy Boyd" in Jerry Maguire (1996). She went through multiple rounds of auditions, but just after she did a screen test with Tom Cruise, she was told that it was down to a choice between her and just one other actress that they had to screen-test: Renée Zellweger (who ended up playing the role).
- Taught aerobics in college and, after arriving in New York City, while waiting to get any acting jobs.
- Has a pit-mix dog named Lucy.
- Started acting at the age of seven when she, her twin sister and their cousin put on plays with names like "Murder on the Sunshine".
- Studied Chinese with Sen. Kirsten Gillibrand, in Beijing, for a summer during university. They both majored in Asian Studies at Dartmouth College. Aisha Tyler also attended Dartmouth College with them.
- Was a theater star in high school.
- Was a sister in Sigma Delta sorority at Dartmouth College, graduating in 1989.
- Has played three characters which all share her first name - Emotional Rescue, The Life Coach (2005) and a guest role in The West Wing (1999).
- Inducted into the Hair Fan's Hall of Fame in 2013.
- Bears a resemblance to actress Jennifer Warren.
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