You didn't specify your configuration. I will assume you have a simple layer-2 ethernet switch and a separate router?
Yes, you can run two independent subnets on the same Ethernet switch. Setup everything on the second subnet using "static" IP configuration (no DHCP, 192.168.2.x/24 subnet from your example above). Do not use DHCP or the DHCP server in your existing configuration will assign addresses in the "live" subnet. Then all of the machines on the secondary subnet will see each other but will not be able to communicate with the 'production' subnet.
Note that this is a very crude approach. It doesn't really create any kind of security boundary between the two subnets. Any machine on the LAN that wanted to listen in in or interfere with the alternate subnet could do it pretty easily by configuring a second address on their network interface. Presuming you have a relatively "trusted" set us users and are just testing it should work fine.
Getting these machines internet access depends on what you are using for a router. If you are using your typical home/SOHO POS router (the thing your ISP gave you or the D-LINK/Linksys/etc thingy from Frys or Best Buy) then you are probably out of luck. Most of them can't do anything outside the simple Single LAN/One Subnet/Single WAN configuration used for simple networks.
If, OTOH, you have any kind of "real" router, just set up the link facing your layer-2 switch to have addresses on both subnets (192.168.1.x and 192.168.2.x). Add a route so that both can see the WAN port with NAT and ensure that there is no "route" set up between the two subnets. Set the "default" route in the static configuration of each host on the secondary subnet to the address you gave to the router (mostly commonly 192.168.2.1).