I imagine the problem will only get worse as bitcoin becomes more popular. It's a bit worrying for the future of bitcoin. Will the cumulative burden eventually overwhelm even hi-spec dedicated bitcoin servers?
Yes and no. Eventually running a full client is going to be like drinking from a firehose, but only professional miners, wallet services, banks, etc are going to bother. Most people will be using thin clients that don't maintain a complete blockchain that is dependent upon a central server to tell it what transactions are important. Much like BitcoinSpinner on Android or Electrum works now.