I have seen this in HEK cells and in primary cultured neurons. It seems to take a repetitive pattern but they can vary in thickness and density. I have seen it on top of and below my cells so I don't think it's something in the plastic.
My colleague also pointed out that there is another fiber-like structure near the bottom of the image. Does this contamination grow or perhaps its some kind of dust?
It appears like dead cells because there is a halo around the cells which generally is seen in dead cells. Moreover, if they do not grow, then they are probably not contamination. These cells are spread above and below your living cells because they were dead at the time of splitting or plating and have continued to stick with the live cells because of improper trituration.