I would agree. I would say with 98% certainty that this is not a database issue or an AC issue. This is purely and simple a networking issue.
While you can connect at times, ruling out OS issues, I would make sure just for fun that File sharing is on and Password protected is off in the Network and Sharing Center. (I am assuming it is Vista since you talk about Start and Network which is not on XP -- you have never told us -- I would find it helpful).
On customize network settings, make sure it is private network and both are named the same.
Now, on a domain, when you reboot, you MUST log in with a user name and password which authenticates you to the server and allows you access to everything on the server which your admin has allowed and it sets up your network connections and reconnects your mapped drives. While mapped drives, e.g. W drive can be useful, they certainly aren't necessary.
The problem as I see it is that every time you reboot your laptop, you are losing your connection to the desktop. Does it ask for a username and password when you log back in? If it does, it is probably only asking a username and password for the local computer (laptop). So, now you are trying to connect to another computer on your WORKGROUP and trying to access a folder that is either password protected (see above) or you do not have credentials for, because you have not logged into it.
At some point you have always mentioned going to Start and Network. When you do this does it ask you for credentials (username and password) to get to the desktop, thereby circumventing the issue? If so, and you do, then that would solve the problem each time.
As George has stated and I have echoed, you may need log on and log off scripts which cancel all mapped drives and re-establish mapped drives or at least re-establish the connection to the database.
I do not think the error message you are getting -- the great big scary box that was shown earlier has anything to do with SQL. I think it is your AC's way of saying your laptop has not shaken hands with your desktop and has no idea where the path is.
I could be way off here, but when it is working, it is working because your laptop is "seeing" the desktop and vice versa. When it is not, it is because your laptop is not "seeing" the desktop. You will probably physically need to log in to the desktop from the laptop. The best way would be to use scripts.
This is done all the time. There are hundreds of thousands of support techs who have to connect to databases where they may work at five sites. They have five batch files, one for each, so they can log in quickly.
I am a bit confused on one thing. How often do you reboot your laptop? It would seem like only once per day. What is wrong with your current situation and workaround?
But, again, just for fun, make sure your network is set to private, the workgroup name is the same and you have file sharing turned on. Then for fun, map the AC folder to the laptop and THEN you should be able to enter the username and password of the desktop and click the button which says Reconnect at log in. I may be all over the place, but you can try.
By the way, are you using Vista or Windows 98?