I'm using 192.168.x.x and have exhausted the IP pool. Since 192.168.0.0/16 is all private, why can't I just change the mask to 255.255.0.0 and start allocating out 192.168.x+1.x IPs?
Edit: I realise that this would preclude me from linking to any other 192.168.x.0/24 networks
Of course you can use a /16, or /23, /22 etc. if you want, I know we have at least a couple of /23's in place and it works fine. Obviously you'll have to alter your netmasks everywhere and especially on your router/s but this is very commonly done.
I personally wouldn't go the whole hog and use a /16, as you say it could cause problems if you're using 192.168.x.y addresses elsewhere, so just choose a /23 or /22 and that should last you a long while.
Nothing stops you doing this - with private address ranges, such as the 192.168.0.0/16 netblock and the 10.0.0.0/8 netblock. Technically you could use non private address space - especially behind an address translating firewall - but it's rude and may become a massive nuisance.
192.16.0.0/24 is a common address range to use. But that's all. People commonly use /24 address blocks, because it neatly maps to the IP address range octets. A netmask is a bit mask that says the first (24 in this case) bits are 'network' and the remainder are 'local subnet'.
That basically means 192.168.0.0/24 is 192.168.0.1 - 192.168.0.255. Another common convention is to adopt '.1' as the gateway, and '.255' as the broadcast. Drop your netmask to 23 bits though, and you double the size of your subnet - 192.168.0.1 - 192.168.1.255. Each bit doubles the size. So whilst a /24 contains 256 addresses (8 'bits worth') - a /16 leaves 16 bits for host - 65,535 addresses.
The same is true of /16 and /8 - they map nicely to the octets of the dotted quad IP format. They are known as class A, class B and class C address ranges and once upon a time, when the internet was a much smaller place, that's all there was. Which is why you'll see a lot of (established) organisations have a /16 or class B, despite - probably - not using most of it on the public Internet.
The whole subnet class thing (A, B, C) is something from the past See wikipedia. It was superseded by CIDR (Classless Inter Domain Routing) Wikipedia.
This means that in "today's" world, you can subnet your private network with whichever network mask that suits you.