Cryptocurrency Private keys... insiders perspective wanted

I just have one question I really need answering.

If I generate a private key for example from myetherwallet.com and store all of my ether at that location, if my ether wallet happens to shutdown for whatever reason, will i be able to access my ethereum from another website or hard wallet with this private key?

I think I need answers for this before i invest in say a Trezor or Nano S.

Any advice would be much appreciated.