Solved. The problem is that there is insufficient time for Simulink to connect to the ADAMS server, so obviously what needs to happen to fix this is to somehow increase that time. The way to do this is to create a new Windows environment variable to override the default value (which is 5 seconds) to a bigger value, like 10 or 20 seconds. To be more specific, here's a step by step on a Windows XP machine:
1.) Go "My Computer", right click and open "Properties" 2.) Go to the "Advanced" tab and click "Environment Variables" 3.) Create a new system variable, and call it ADAMS_CONTROLS_WTIME 4.) Set the value to something greater than the default, which is 5 (unit of time is seconds). The greater the value, the more likely it is that you will get a connection and not get an error, but the longer of a time you will have to wait for each and every simulation to run. 10 seconds seems to work best for me. 5.) Click "OK" a bunch of times, then restart MATLAB. You may need to reboot your whole machine, I'm not sure.
Hopefully that does it! It seems to have fixed it for me.