Public Key? Where would you even get that?
Private key? No! You should not be typing the private key into a public website. The private key can be used to steal/spend your bitcoins. A partial private key won't tell you ANYTHING about the bitcoin address. This is just a bad idea all the way around!
Public adress? You mean the bitcoin address? Ok, now we are getting somewhere...
If you type the first few characters of a bitcoin address, you will get the first address that ever received bitcoins that uses those starting characters. Sometimes (such as with 19QkqAza) this will result in the address you are attempting to look up, but that is not guaranteed. Multiple different addresses could start with the same characters, but only one of them will show up when you search with those characters. The one that show up might not be the one you are looking for. It is a very bad idea to rely on firstbits as a way of finding an address. The fact that it works occasionally is not proof that it works.
Consider the following question:
"Can I identify an animal as a dog if it has fur?"
Then consider the following proof:
"Look right there, I'm pointing to an animal with fur and it is a dog"
Does this mean that I can identify an animal as a dog if it has fur?
You can't use limited examples as proof of your assertion.