I have seen a number of arguments for the existence of God that boil down to the following:
1) Fact X doesn't rule out a weak form of Deism.
2) Therefore God exists.
Now there are many circumstances in which (1) is true - indeed there are few facts that will rule out a weak form of Deism.
My observations are not inconsistent with Russell's teapot and the invisible pink unicorn in my garage either; that doesn't make them reasonable positions. And not impossible is a very, very long way from true... or even "seems likely".
But in any case, if someone wants to take a Deist position, we will probably have little to argue about. Or if they want to believe in orbiting spouted ceramic containers of steaming Orange Pekoe, that's no terribly great loss either.
However, the conclusion simply doesn't follow, even if the God of (1) and the God of (2) were the same God. In practice, they never are. Indeed, the argument is more like:
1') Fact X doesn't rule out an invisible pink unicorn in my garage.
2') Therefore JFK was a reptoid.
Not only is it not a valid syllogism, it's a total non-sequitur. Sentences (1) and (2) are completely unrelated, because they contain no common elements.
This argument is well disguised by using the term "God" twice. It's a very neat bait-and-switch con - just like a used car dealer, the car that got you in and the car you were actually sold can both be described as cars... but they're NOT the same object.
And so with this argument - it's just as much of a con.
The God of (1) is far from the God of (2).
Indeed, often on closer examination, the God actually being introduced in (2) turns out to be inconsistent with the fact in (1), but that's of no concern to the person making the argument - once you've accepted proposition (2), standard apologetics take care of that little problem in (1)!
Sentence (2)'s God has been snuck in through the back door, and you're supposed to uncritically accept all the attributes of God (2) - based on an argument that God (1) isn't completely impossible.
Watch out for it. It's interesting how often some form of it crops up.