Is this kind of network possible?

SPEED.

Nothing you noted above will increase or add more speed to your internet connection.

Is the setup wrong? No. It looks like a normal setup and about all you can do is to ban those that run TORRENTS on your network. If you feel it's OK then all I have ever needed to do is wait till the new network person figures this out.
Bob

Having multiple IP addresses available to use is not the same as having multiple internet connections. You won't be able to exceed your 5Mb speed cap unless you add another connection or increase the speed of your current connection. (Talk to your ISP about this.) 20 PC's all sharing a single 5Mb connection for anything that requires low lag is a recipe for broken hearts.

The recommendation to more strictly control your internet traffic will probably solve most of your problems. There's no end to hardware and software solutions for this, but getting yourself a good hardware firewall and banning torrents and other bandwidth hogging activities (HD YouTube for instance) will pretty much do the trick.

Also, if you have wireless, make sure your network is locked down tight and isn't being leeched by other homes/businesses nearby. One of the things you can do is log all your internet traffic and see who's doing what, as well as checking the DHCP leases in your router to see who's connected and how long they've been there.

With most equipment and software, you can reserve a portion of your bandwidth for various programs/ports and thereby guarantee that your critical network traffic is unobstructed.

Finally, make sure that all your DHCP leasing is happening on the same device, and that its the fastest device in your lineup to minimize the potential for conflicts and streamline the network overhead.