I'm posting this here because I have not found this documented anywhere, so others might find it useful.
The scenario is as follows: an OLAP connection is created to a BW system, and can be confirmed to work via the CMC (using "Choose cube") or via WebI Desktop. However, when one tries to use the browser-based WebI, the error below is thrown:
This error has several causes apparently, judging from Notes 2088944 and 2062091. In these cases, the cause is a symbol in the password of the BW account. That was not the case here, as the password did not contain any symbols. Other notes suggested a BW password change may cause this as well, because the relevant APS server caches the password originally used and the new one is not forwarded to BW. We have had that very same issue just hours before a Demo and that can ruin your day 
. Another cause to check is authorisation in both BW (of the BW account to the cube) and BI (authorisation of the user to the OLAP connection).
Now back to the issue at hand. The logs will "help" you. The WebI processing server GLF will claim that the "password logon is no longer possible" because of "too many failed attempts" on the BW server:
Well, you know the password works, there is no authorisation issue on either sides, no symbols in the BW password, so what is wrong? Another note will suggest restarting servers, SIA and even re-creating APS servers when the BW password is changed. We tried all that, but in this case, the password had not changed, so re-creating servers, etc., did not help.
In the end, the problem is that there was an invalid configuration for SAP authentication in the area CMC > Authentication > SAP. An invalid "Entitlement Systems" entry was for some reason causing the problem. When clicking on the tab "Role Import", the same error "too many attempts" would be thrown.
Interestingly, all OLAP connections were using Pre-defined Authentication (not SSO), so one would think SSO SAP authentication would not affect these connections. Well, think again.
Solution: in the end, we just removed the offending SAP Entitlement records (there were two), restarted the APS and WebI processing servers, and the connection works. Hope this will help someone.