Not really. Your yeasts actually depend on the small amount of oxygen in the headspace to multiply and convert the priming sugar into CO2 and a little more alcohol to carbonate up your bottles. This headspace then quickly fills with CO2 gas as a byproduct of priming sugar fermentation. The pressure of the CO2 that builds in the space forces more CO2 into liquid suspension, so having just a small amount of headspace is somewhat desirable. I use longneck bottles and fill mine to the top with a bottling wand. When I pull the wand out of the bottle after it's full, I have about 1" of headspace in the neck which is just about perfect.
The only exception to this is when you force carbonate using compressed CO2. These beers don't depend on any bottle conditioning to carbonate the beer as CO2 is pressurized into them using a CO2 tank and a regulator. This is typically done in a keg first under moderately high CO2 pressure and then beer is filled into individual bottles for sharing/judging/etc. using something along the lines of a beer gun (or bottle filler in a commercial brewery). These beers are best capped "on foam," meaning the bottles should be filled completely to the point of foam dripping down the sides to eliminate any headspace in the bottle.
The reason for this is oxygen exposure on beer that's already carbonated just promotes oxidation, as there isn't any priming sugar for yeast to consume the remaining oxygen. So, they fill the bottles and kegs as full as they can.