Hi All,
These two vertices are are right next to one another. I have given them a little more space for the purpose of the image.
Why isn't merge by distance working here? 
For those of you that have this issue. When I clicked merge by distance there is a settings panel that appears in the bottom left hand corner of the screen 
It was set at like .0001 and it was not merging I just increase this value on the slider arbitrarily to see what happened and it seemed to expand the radius enough to merge the vertices. The value in the image I think is a bit high, but like I said just did it to see what happened. I think just a little higher than the .0001 would have done the trick.
Thanks All!
Normally the default value is enough for the merge by distance parameter to work, because in order for the vertices to be merged, it is best if they are occupying the same space in all three axis. So getting them really really one on top of the other so that they are basically occupying the same space.
Sometimes you can be looking at the vertices from an angle that you don't realize they are far apart and you only realize it when you turn the camera around, but you didn't realize it because they were occluding themselves. When in that situation, you try merging by distance and it doesn't work and it leaves you scratching your head. Again, you have to make sure the vertices are really really close.
The distance parameter, the one you used fiddling with the numbers, comes handy in those sorts of situations, because the merging of the selected vertices is triggered by how close than a specified distance they are to each other. So if they are 0.05 distance apart, you tell Blender that anything that is closer than that will be merged, anything more apart, so if it's 0.06 distance apart will be ignored and won't get merged. When you increase the value, the gap in distance of what gets merged gets wider, and if vertices fall into that range, they get sucked in like a black whole.
