192.168.x.x addresses with subnet 255.255.0.0

We are designing a network where we have

192.168.100.52
192.168.110.13

Initially the subnet mask was 255.255.255.0, so clearly these first two machines will not communicate with each other without some sort of router.

Is it legal to change the subnet mask to 255.255.0.0 to allow these two networks to be part of the same network?

The 192.168.* range is private so you can technically do this without causing IP address collision problems on the wider Internet. However, conventionally the 192 octet range is used for Class C networks, which have a subnet mask of 255.255.255.0, allowing communication across only one octet range of computers (255). What you are describing is a Class B network, which by convention uses 172.16.x.x

Convention aside, there is nothing stopping you from using the 192.168 range as a class B. If you wanted to limit network traffic to only the 192.168.100.x and 192.168.110.x networks, you could change the subnet mask to something that includes only that range, such as 255.255.240.0 (which would allow communication between any subnet in the range 192.168.96.1 - 191.168.111.254) - if you put the two subnets together (192.168.100.x and 192.168.101.x) you could change the subnet mask to 255.255.254.0.

A fun Subnet calculator to play with can be found at http://www.subnet-calculator.com/

Yes. The 192.168.x.x range is B class private range so you can break it down into as many (or few) subnets as you like. Using 255.255.0.0 as a subnet mask is fine. This will make one big range with 65534 IP addresses (that's 256 * 256 - 2 IP addresses; 192.168.255.255 is the broadcast address and 192.168.0.0 is the network address both of which can't be assigned to devices on the network).