The thing that fixed it for me was to upgrade the SDK platform-tools to r16.0.1. For me, this version was not displayed in my SDK Manager, so I pulled it down from http://dl.google.com/android/repository/platform-tools_r16.0.1-windows.zip directly.
You then need to rename the
platform-tools directory and unzip it to
android-sdk-windows/platform-tools. Using the SDK Manager, I had also updated to the latest sdk-tools before this.
If your whole Eclipse and ADT are ancient, you may need to update them as well, but I didn't need to.
Note: you may need to run SDK Manager twice (once to update itself) before you will see the latest packages.