I actually just gave a presentation on anti-inflammatory treatment and impaired muscle adaptation. NSAIDs (i.e. ibuprofen) blunt the immune response caused by inflammation. Working out the muslce, especially with heavy eccentric contractions, causes muscular damage. One side effect of this is inflammation of the damaged tissue, caused by neutrophils and macrophages (immune system cells). Inhibiting this immune response is presumed to either partially inhibit or slow the repair of the damged muscle.
The two main macrophages looked at in most studies are ED1+ and ED2+. ED1+ is responsible for clearing the necrophilic (dead/damaged) tissue and proliferating type 1 collagen production (which is used for structural integrity of the muscle). ED2+ is responsible for stimulating the action of satellite cells, which repair the damaged muscle fibers. Inhibiting these factors using anti-inflammatory agents thus inhibits the muscular adaptation to exercise.
It is also a dose- and time-dependent response, meaning that the higher the dose and the longer you take a certain dose the more impaired this adaptation is.
As far as protein synthesis being inhibited, while I have not directly looked into it myself, it has come up in a lot of the research I've come across.
Taking high doses of anti-inflammatory agents is fine as it will help with any injury you may have. After the inflammation is gone (typically treatment is ten days) you can resume training, at least in a rehabilitation phase if not healed. It would be better, IMO, to locally inject these agent so you don't have a systemic response to them. Chronic ingestion of NSAIDs is not recommended unless you absolutely need them.
You should also still train while taking anti-inflammatory agents, as there will still be an adaptive response. During the NSAID treatment phase I would recommend focusing on concentric contractions, which cause less muscle damage than eccentric contractions.