Bonney was born in Montclair, New Jersey. As a child she studied piano and cello. When Bonney was 13 her family moved to Maine, where she became part of the Portland Youth Orchestra as a cellist. She spent two years at the University of New Hampshire (UNH) studying German and music including voice with Patricia Stedry, and spent her junior year at the University of Salzburg, where she switched from cello to voice. While there, she studied at the Mozarteum. Years later she received an honorary doctorate from UNH.
In 1979, Bonney joined the Darmstadt Opera, where she made her debut as Anna in Die lustigen Weiber von Windsor (The Merry Wives of Windsor). In the subsequent five years she made appearances throughout Germany and Europe, notably at the Royal Opera House Covent Garden in London and La Scala in Milan. She made her Metropolitan Opera debut in 1987 in Richard Strauss's Ariadne auf Naxos in the role of Nyade and her Vienna Staatsoper debut the same year as Sophie in Der Rosenkavalier. Since then, she has appeared at the major opera houses of the world and at the Salzburg Festival, where she was Servilia in Mozart's La clemenza di Tito. Her reprisal of this role in 2006 was cancelled.
Along with her wide-ranging repertoire in opera as a lyric soprano, she is a distinguished recitalist in command of many languages. She appears on more than 90 recordings, including 15 solo recitals. For two years, starting in 1999, Bonney did not perform in opera, to focus on lieder recitals. However, she noted that solo recitals lacked the camaraderie of performing in an opera production with many other people.
On August 1, 2006, IMG Artists announced that all forthcoming appearances by Bonney were cancelled and that they would no longer be representing her. At the time, the IMG Artists website only stated the reason as "due to personal circumstances", but Bonney has stated in a July 2007 article that those circumstances included her "difficult" divorce from Maurice Whitaker. On March 16, 2007, the Rheingau Music Festival announced that Bonney would resume her career on August 16, 2007 by standing in for the indisposed Thomas Hampson. Before that scheduled concert, on July 22, 2007, Bonney stepped in as one of several vocal soloists to replace Renée Fleming, when Fleming canceled her appearance at a Verbier Festival concert.
In 2006, Bonney recorded the "singspiel/dramatic oratorio" Welcome to the Voice, composed by Steve Nieve, with Sting, Amanda Roocroft, Robert Wyatt, Nathalie Manfrino, Sara Fulgoni, Elvis Costello and Brodsky Quartet. This is part of her resumption of concert and musical activities, which includes new management representation by Michael Storrs Music Limited.
Bonney has been married and divorced three times, first to the Austrian tenor Michael Roider, second to the Swedish baritone Håkan Hagegård, and third to the British music management executive Maurice Whitaker.