Why is it that a person can't take a large amount of over the counter pain relievers, like Tylenol or Ibuprofen, to get the same effects as stronger painkillers like Vicodin or Hydrocodone?

Different method of action. Opioids like hydrocodone and Oxy work by binding to neuroreceptors(neuro means brain) which send a depressive signal, which basically means that the signal numbs your central nervous system.
I don't remember the exact MoA of the others so to just put it simply, ibuprofen and aspirin work by reducing inflammation in your body, and Tylenol works by simply cutting off the pain signal on its way to your brain.