Actually, I think the problem is that she tries to sing those notes strongly when for her voice type they would sound a lot better sung more quietly and delicately. On one of the newest recordings we have of Home and Fools Like Me, she does a better job of hitting the notes and using a bit of vibrato instead of just the flat note, but I still think she could take a lesson from Jewel or something on how to do high notes. Sarah McLachlan does a good job of singing the high notes strongly (listen to "Fear"- great song), but her voice is a lot different than Vanessa's and she has had a lot of vocal training.
And I've always wanted to cover "Fear", too. I'm a soprano but my voice has gotten a little bit lower recently and I can't always hit the part - "there's nothing I'd like better than to fall" without sqeaking. Hehe. I found a reallllly good recent live performance of the song. As unbelievable as it sounds she actually sounds better singing it live than on the studio version and it's like 10 years later!