J.-L. Lions proved several observability theorems for coupled linear distributed systems provided the coupling parameters are small enough. It remained an open question whether the assumption on the smallness of these parameters is necessary for the validity of the results. Using nonharmonic analysis the first two authors proved recently that in some cases the observability holds in fact for almost all values of the parameters if the underlying domain is an open ball. Combining this method with a compactness-uniqueness method developed by the third author we extend these results for all sufficiently regular bounded domains.