I believe it has to do with stress. In a simple sentence like S+V+O, the clausal adverbial (inte in this case) immediately follows the verb, as you probably understand. But if one chooses to unstress the object, it can go before the clausal adverbial. So...
Jag såg honom inte. I did not see him (stress on I... maybe others saw him)
Jag såg inte honom. I did not see him (stress on him... I saw others, but not him)
Jag känner inte honom. I don't know him (stress on him... I don't know Erik, but I know others people)
Jag känner honom inte. I don't know him (stress on I... I don't know Erik, maybe others do)
I don't know how well that was explained, so it'd be best to wait for a native.
Jag såg honom inte. I did not see him (stress on I... maybe others saw him)
Jag såg inte honom. I did not see him (stress on him... I saw others, but not him)
Jag känner inte honom. I don't know him (stress on him... I don't know Erik, but I know others people)
Jag känner honom inte. I don't know him (stress on I... I don't know Erik, maybe others do)
I don't know how well that was explained, so it'd be best to wait for a native.
I second everything Halfdan said, with a minor remark. With standard word order, as displayed in (1) and (4), the "stress", if any, is on the verb. If you want to stress that you, as opposed to everyone else, didn't see him, you simply have to stress that you, which is not necessarily standard, although Halfdan's examples might indicate this. Do note that this distinction cannot smoothly be made in writing.
A few examples, where (1) is standard and the other two simply variations, to further clarify:
(1) Jag såg honom inte. - No stress. Was he there? Well, you didn't see him.
(3) Jag såg honom inte. - Stress on jag. He might have been there, but you didn't see him.
(2) Jag såg inte honom. - Stress on honom. You didn't see him, but yes, you saw others.
A few examples, where (1) is standard and the other two simply variations, to further clarify:
(1) Jag såg honom inte. - No stress. Was he there? Well, you didn't see him.
(3) Jag såg honom inte. - Stress on jag. He might have been there, but you didn't see him.
(2) Jag såg inte honom. - Stress on honom. You didn't see him, but yes, you saw others.
On a side-note, many people fail to distinguish between "han" and "honom", so I think you often hear "jag känner han inte" (which is wrong of course!) 
On the sidenote: as jackuppsararen says poeple are sometimes ignorant.
But this also happens in contracted speech: "ja känner'an inte".
It's just too difficult to contract "honom".
And as a consequence nobody feels much guilt in doing it wrong this way, though you do recognize that you're not speaking rikssvenska anymore.
Or maybe it's the other way around - no rikssvenska, no guilt .
But - learn the correct way first.
But this also happens in contracted speech: "ja känner'an inte".
It's just too difficult to contract "honom".
And as a consequence nobody feels much guilt in doing it wrong this way, though you do recognize that you're not speaking rikssvenska anymore.
Or maybe it's the other way around - no rikssvenska, no guilt
.But - learn the correct way first.
Bolded by me.On a side-note, many people fail to distinguish between "han" and "honom", so I think you often hear "jag känner han inte" (which is wrong of course!)
Quite correct in regional Göteborg dialect. Here, among many, the historically totally correct ackusative han still lives on. Honom is the old dative, which for some obscure reasons has crowded out that han from standard Swedish.
Please do, but don't automatically link object han to poor language.