I think people often forget that inflammation isn't (usually) the evil at hand, but rather the body's response in fighting whatever it may be.
Research is still somewhat still up in the air but most signs point to the fact that NSAIDs will either delay healing, or perhaps even reduce healing musculoskeletal injuries.
For instance a study in 2013 found that NSAIDs impaired bone fracture healing and tendon to bone healing, while suggesting that they may benefit tendon specific healing. However they did suggest that there are exceptions to the tendon healing findings and that the larger clinical picture of a patient should be considered. (http://jap.physiology.org/content/115/6/892.long)
A 2009 report found that NSAIDs (in regards to tendon healing) are logically a double edged sword. NSAIDs work by inhibiting an enzyme that synthesizes inflammatory mediators. They may benefit tendon repair by influencing how the tissue remodels. However, the mediators NSAIDs act upon are vital to the early tendon healing response as they control blood flow to the area, and normal tissue remodeling may be impaired.
I could go on but basically what I'm getting at is (and as most science goes); it depends on a lot of factors you would have to look at a patient specific level. You have to ask is it acute? Is it chronic? What exactly is the underlying cause of the inflammation? Is that treatable?
They are symptom specific treatments. Is it going to hurt to use them for some relief for a minor injury? Probably not. Would you be better off without using them in that scenario? It seems that way. Could people use them to mask issues they should probably seek help for? Certainly.
The actual problem needs to be identified and treatment needs to be tailored. In most cases the body is doing its own thing and with good cause, short of autoimmunity and cytokine storms.