Please take a look at the following example:
Let's see if we can't get there before they get back.
I'm pretty sure the sentence is grammatical and its semantic is quite similar to:
Let's see if we can get there before they get back.
But then what's the point of using the first version?
In my language the situation is similar and we use it as the speaker is giving the sentence extra emphasis by using the negative form.
Consequently, by checking if the opposite of one thing is impossible then the thing we want to do must be possible.
What about in English? Please let me know your thoughts on this.