Some things to try....
If you have a single broker for multiple accounts, when selecting to add another account for the same broker: Clear cookies and then search for your broker via the search bar rather than clicking the broker icon. This should allow the screen to input account information and import the data.
For Fidelity this user found a way here....
After I had downloaded from one account and the second one would not work, I looked at the Fidelity website and noticed that I was already logged in to my account (the one I had downloaded from). Turns out that TurboTax had not dropped the connection from the first account. I logged out of that one and tried to download the second account and I was presented with the login screen and could then download from the second account.
