You're just not getting it.
When you input address and netmask for network configuration for an interface, networking stack will calculate all the possible addresses that belong to that network. Meaning input 10.0.0.1/24, and OS will mark down IPs 10.0.0.1-254 as broadcast domain, hosts that are directly reachable over layer 2 without routing.
So input 192.168.0.1/8, and OS will think that all IPs DutchDaemon mentioned are directly reachable.
If you have IP 192.168.0.10 and netmask 255.255.0.0, and you try to connect to 192.1.122.17, OS will see that IP doesn't belong to the calculated reachable hosts, and forward those packets to appropriate gateway. On the other hand, if your netmask is 255.0.0.0, that IP becomes part of the whole subnet, meaning OS thinks it's directly reachable. So it won't forward packets to NAT gateway, it'll broadcast an ARP-WHOHAS over layer 2. Normally, no-one is going to answer, since that host is out of your layer 2 domain. And you've effectively severed communications between that box and 16 million Internet IPs.
This whole thing doesn't have anything to do with NAT, it's basic network addressing. Yes, your router won't send packets with private IP originators to WAN, and yes, if it does, ISP will filter them out, but those packets never reach NAT because system thinks they are inside a local network, not destined for outside.
I am sorry if I sound rude, but if you still can't grasp this, then please take two boxes, connect them via crossover cable, configure one as 192.168.0.40 / 255.255.255.0, other 192.168.0.20 / 255.255.255.224. Try to ping between them. I never had reason to do so, but I can predict that you're going to get a "no route to host" error. Because of the netmask, valid range of IPs would be 192.168.0.1 to 192.168.0.30. System will think that destined IP doesn't belong to local network and will try to route the stuff through default gateway, which in this case isn't even configured, so it won't have a route. That's the opposite example, where you cut out a LAN host because of a wrong netmask. In this case, system thinks that's somewhere out, while in original example we're talking about, system thinks that Internet destination is somewhere inside.
When you input address and netmask for network configuration for an interface, networking stack will calculate all the possible addresses that belong to that network. Meaning input 10.0.0.1/24, and OS will mark down IPs 10.0.0.1-254 as broadcast domain, hosts that are directly reachable over layer 2 without routing.
So input 192.168.0.1/8, and OS will think that all IPs DutchDaemon mentioned are directly reachable.
If you have IP 192.168.0.10 and netmask 255.255.0.0, and you try to connect to 192.1.122.17, OS will see that IP doesn't belong to the calculated reachable hosts, and forward those packets to appropriate gateway. On the other hand, if your netmask is 255.0.0.0, that IP becomes part of the whole subnet, meaning OS thinks it's directly reachable. So it won't forward packets to NAT gateway, it'll broadcast an ARP-WHOHAS over layer 2. Normally, no-one is going to answer, since that host is out of your layer 2 domain. And you've effectively severed communications between that box and 16 million Internet IPs.
This whole thing doesn't have anything to do with NAT, it's basic network addressing. Yes, your router won't send packets with private IP originators to WAN, and yes, if it does, ISP will filter them out, but those packets never reach NAT because system thinks they are inside a local network, not destined for outside.
I am sorry if I sound rude, but if you still can't grasp this, then please take two boxes, connect them via crossover cable, configure one as 192.168.0.40 / 255.255.255.0, other 192.168.0.20 / 255.255.255.224. Try to ping between them. I never had reason to do so, but I can predict that you're going to get a "no route to host" error. Because of the netmask, valid range of IPs would be 192.168.0.1 to 192.168.0.30. System will think that destined IP doesn't belong to local network and will try to route the stuff through default gateway, which in this case isn't even configured, so it won't have a route. That's the opposite example, where you cut out a LAN host because of a wrong netmask. In this case, system thinks that's somewhere out, while in original example we're talking about, system thinks that Internet destination is somewhere inside.