There have been a number of studies over the years that show Caffeine and Acetaminophen (Tylenol) combined is more effective at pain management than any of the common OTC pain relievers and could be used in place of narcotics in some circumstances.

Mixing Caffeine, Acetaminophen and Aspirin has been around for a LONG time, Excedrin for example, that works well on headaches especially migraines.

Recently a study came out that shows Acetaminophen (600mg), Ibuprofen (400mg) and Caffeine (15mg) work well in managing pain post molar removal, but the study didn't compare it directly to any narcotic pain medications.

In other countries you can buy Acetaminophen (325mg), Ibuprofen (200mg) and Caffeine (40mg) combined OTC and the FDA in the US just approved Ibuprofen/Acetaminophen mix OTC.

If there is a study that show Acetaminophen/Ibuprofen/Caffeine to be as effective as Narcotics I haven't been able to find one. There is a lot of anecdotal evidence that they work well and appears to be safe as long as you can tolerate them individually... Follow the recommended doses/timing of each of them individually.