All cars do not have the same fuel systems, so we'll take a very generalised stab at this.
Running the car very low on gas could damage the fuel pump on many modern vehicles which have an in-tank fuel pump. The fuel acts as lubrication for the pump, and running it too low can cause the pump to burn out. Older carbureted designs tend to have mechanical pumps that operate at much lower pressures than new electric ones. These are typically less likely to be damaged by a low fuel level.
Many modern cars have internal fuel filters. Instead of replacing these filters every so often as you would an oil filter, they are "cleaned" (for lack of a better word) by the unused fuel returning back through the filter. If this is a newer vehichle, I would not expect this system to fail easily. from personal experience, it tends to be fairly robust. If it is 7 or 8 years old and the filter has never been touched, it is certainly more likely to have failed. If you have been putting sketchy gas into the vehicle, it is certainly more likely to have failed. If it is an old cannister type filter, might as well just slap a new one on; they are very inexpensive.
Not all of the generalizations here apply to all cars in the "modern" and "older" categories.
Certain makes and models of vechicles are more likely to have an issue. As an example only (you'll have to look into recalls for you make and model), here is a recall from 3 months ago for BMW's with failing fuel pumps. I have a 3 series that is on its 3rd, thankfully warrantied, fuel pump.
Given that the issue did not start until 5 days after almost running dry, I don't think it is safe to say that they are definitively related. Many other factors could be affecting the situation and giving you similar symptoms.