Fidelity Investments is listed as a TurboTax 1099 partner and therefore the import should be supported. However, there are some situations where the import might not work. It could be something simple like selecting a slightly different brokerage name, using an unstable internet connection, or maybe using a different account number format.
Another solution is to log into your Fidelity account when prompted using a different browser as sometimes this also could be an issue if a pop up blocker is installed.
Please see "What if I can't import my 1099" for more suggested solutions.