Create a Linked-Server from SQL Server 2008 R2 to AWS RDS SQL Server
On SQL Server 2008 R2 the default value of remote login timeout (s) is five (5) seconds, I believe, though it is configurable.

Since you are now trying to connect to an AWS RDS SQL Server, have you considered whether the remote login timeout (s) is appropriate for this environment?

Try raising the timeout a bit. You might start with 10 seconds:

sp_configure 'remote login timeout (s)',10;
RECONFIGURE;


I have had to bump up the timeout seconds in some environments, so do not be afraid to try this.  (It is equally simple to set it back to the previous default.)  

You can try to configure a longer timeout, if needed.  So far 10 seconds has almost always worked for my environments, but I have gone to 20 seconds a couple of times.