Other servers in Finland worked fine (I have been able to use eduroam the whole time with my TUT accounts) and only difference we first found was that Helsinki University certificate was bigger than the ones in some of the working organisations. This lead us to suspect RADIUS message fragmentation problems somewhere in between European Top Level RADIUS and Matti's device. We then remembered the one Radiator configuration directive, which we found was missing from Helsinki University RADIUS server configuration.
Adding EAPTLS_MaxFragmentSize 1024 solved this problem so if you are having similar problems with eduroam and your home organisation is running Radiator RADIUS, we suggest checking at least this setting found in Radiator manual:
For TLS based EAP types such as TLS, TTLS and PEAP, this optional parameter specifies the maximum size in octets permitted for each TLS message fragment. Defaults to 2048, but many EAP clients, routers and wireless Access Points have limitations that require EAPTLS_MaxFragmentSize to be set as low as 1000 or less. Setting this number too small can result in excessive RADIUS request round trips during EAP TLS authentication, slowing down the authentication process. Setting this number too large can result in failure to complete TLS authentication for some types of clients and devices.