This typically happens when the system of units of the grid is changed between metric and imperial during routing. Even with very fine sub-divisions, it is quite impossible to align the endpoints of two tracks drawn in different units exactly.
One behavior of EAGLE is that when you start a new track, the start-point snaps to the next position of an air wire (you need to hold SHIFT to let a track start just under your mouse pointer), but the end point will never snap to anything than the grid.
There's just one hint that you indeed hit the correct position: There's no new line segment sticking to your mouse pointer.
Due to this, I'd recommend to stick with one unit for the grid, changing not more than the number of sub-divisions.
If a part does not fit onto the grid, don't route tracks into the pads of this part, as they will not snap to the precise positions of the pads. Instead, route tracks out of the pads, since then they will snap to the correct position, and after the next kink, the track is on-grid again.
Since you are convinced the board is OK, you could just send the board into production, but this could lead to queries and delays. And as said in the comments, you could get unexpected results if for example the line ending type changes somewhere during the conversion from EAGLE format to whatever the manufacturer uses.
So, it's better to get rid of the airwires, and I'd suggest to let the auto-router do the job.
Be a little careful, the auto-router will not change its track with to match the width of the existing stubs. If it uses a too large width, you get knobs in your tracks, which can lead to clearance errors. On the other side, if the stubs don't overlap, you can get too thin track segments.