Kerberos Authentication Issue
Definitely sounds like a double hop (Kerberos) issue. When you're rdp'd into the box, then the cube is only 1 hop away so I'd guess that's why it works. 

Start a profiler trace on the analysis services server, then query the linked server. Check the NTUsername field in the trace. If Kerberos Delegation is failing, the NTUsername will be something like "ANONYMOUS LOGON".

From Server1, what does the following return?

select auth_scheme from sys.dm_exec_connections where session_id=@@spid


What Delegation permissions did you set up in AD for the machines?

Does the spn match the server name you are trying to connect to? If you have alias's, cnames, etc, you'll need an spn for each that you wish to connect to via kerberos.

Also, try out the Kerberos Configuration Manager to verify your settings. This can be particularly effective for reviewing what SPNs actually exist.