Other mistake: In the beginning, Adaline's husband dies. At the funeral, their daughter looks to be about ten. But then when Adaline is driving to her parent's place and the commentary says, "Where her five-year-old daughter is waiting for her." There's no way the child at the funeral was five years old.
Other mistake: Near the end, Ellen Burstyn is holding the new puppy, referring to the dog as a female. Right after that, she is talking to Ellis with the puppy there, and they refer to the dog as a male.
Chosen answer: It was because she had used so many fake I.D. the F.B.I. had become suspicious, possibly believing she was involved in some type of criminal activity, which is often why people assume different identities.
raywest ★
No, that can't be right because she doesn't start using fake names until AFTER the incident with the FBI. She was still using her normal name, Adeline, when the FBI grabbed her. The implication of the story is that they grabbed her because she didn't age and they wanted to use her as a lab rat.