It depends on the version (of Windows 7 Home premium) you bought. If you purchased an OEM version (which should only be purchased with a computer), it is only valid for that one computer and the licence does not allow you to move it from one machine to another.
If you purchased a retail edition eg a normal boxed version from a shop, the licence states that you can use it on one computer at a time.
So, provided that it is a retail version you want to move, it sounds like you are not going to act against the terms of the licence.
As to whether or not you have trouble activating the licence, that is a different question. Both retail and OEM versions will activate on more than one machine, but there are limits on how often each will activate before Microsoft's servers get the hump with you.
If you do end up talking to a Microsoft employee, if you tell him or her that an OEM version is going on the same machine they will let you activate it. If it is a retail version Microsoft will just want to check that you are only installing it on one machine.
Edit: Looking at (well, quickly scanning through) the licences from Microsoft, it seems that the OEM licence, not longer states that it cannot be transferred from one computer that you own to another that you own. You are not allowed to have it on more than one computer and you cannot transfer your rights except for with the machine it relates to. Maybe I missed something...