Addition of third computer in network causes router to drop internet connection (is the problem Dropbox???)

Question

This is not specifically about MacBook Pros, per se, but since all of the communities are based on specific products, I chose it since it applies to two of the devices concerned.
We have a LINKSYS WRT54G router in our home connected to cable internet.
We normally run a number of iOS devices on it at any given time (iPhone 5, iPhone 4s, iPhone 4, two iPad 2nd gen., one iPad Mini with Retina), and almost at all times, an Intel MacBook Pro on 10.6. We sometimes also have an Intel MacBook (black) on 10.6. These all live in harmony just fine. Both laptops have Dropbox on them, but signed in on different user accounts.
I used to have someone working at my house quite often on an Intel MacBook Pro on 10.7, also using Dropbox (but signed into a unique account). Whenever she had her computer on in our house, it would often cause the network to just drop the internet connection suddenly. The network would still have a strong signal, internet just wouldn't work.
We never quite figured it out. We had to reset the router constantly. It was NOT the modem -- you could reset just the router and the results were the same as if you had reset the modem. I didn't pursue a fix since she has since stopped working here.
Recently we just added an Intel iMac on 10.6 to the house, and it is doing the same thing. The only difference is this time, it's logged into the same Dropbox account as my MacBook Pro. At first, it seemed as if there was an issue if both were trying to update at the same time. When I paused syncing on one computer or throttled the upload/download bandwidth, it seemed to get better. However, lately, I am not syncing any large files -- or any files at all, it seems, sometimes -- and I also kept one machine on throttled connections, while a THIRD machine is even off completely...and it's still kicking the internet connection off.
Any ideas? I'm getting really tired of re-setting it every so often.
Thank you!

Answer 1

Also, there are no other networks we pick up here. Sometimes, we randomly get the neighbor's, but it's protected and we don't ever join it.
Try the basics first:
   Change your router channel number.  Most times this works & is all you have to do.
Disconnect & reconnect your modem.  unplug it for about 10 seconds.  Plug it back in.  Do the same for Apple’s routers.  Wait for everything to reboot.
System Preferences>Network
Click the Assist Me button.
In the next window that pops up, click the Diagnostic button & do the necessary.
Research Knowledge Base for network problems that pertain to the OS that is currently installed on your computer.   See these basic networking KB Articles:  http://support.apple.com/kb/HT1401 AirPort troubleshooting guide
http://support.apple.com/kb/HT4628  Wi-Fi: How to troubleshoot Wi-Fi connectivity
http://support.apple.com/kb/HT2712 Using network locations in Mac OS X
Manually provided DNS server addresses are higher priority than DHCP's
http://support.apple.com/kb/HT1714 Solutions for connecting to the Internet, setting up a small network, and troubleshooting
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What to do when you can't connect to the Internet
Also, run the Airport Utility app which is located inside the Utilities folder.
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If using a  Linksys router, contact LinkSys Customer Support and/or post in their forums.
If using Apple's Airport, please re-post over in one of the AirPort Forums.