Yep - I work at a large consulting company, and everyone has their own Amex. It's a corporate issued Amex, but it's your name on the card and you're responsible for paying it off. The benefit is, you get a listing of transactions in our expense software, and I can link specific transactions to specific clients or jobs. Then the reimbursed expense shows up in your bank account. Or, I can mark them as personal and pay for them myself.
I used to work for one of the big consulting firms. We were required to use a corporate Amex but our expense software wasn't integrated, so we still had to copy over every single transaction manually.