Can science test the supernatural? Yes!! is an interesting article about why there are not "two nonoverlapping magisteria". The core of the article is based on the paper Can Science Test Supernatural Worldviews? Abstract of the latter paper:

Several prominent scientists, philosophers, and scientific institutions have argued that science cannot test supernatural worldviews on the grounds that (1) science presupposes a naturalistic worldview (Naturalism) or that (2) claims involving supernatural phenomena are inherently beyond the scope of scientific investigation. The present paper argues that these assumptions are questionable and that indeed science can test supernatural claims. While scientific evidence may ultimately support a naturalistic worldview, science does not presuppose Naturalism as an a priori commitment, and supernatural claims are amenable to scientific evaluation. This conclusion challenges the rationale behind a recent judicial ruling in the United States concerning the teaching of “Intelligent Design” in public schools as an alternative to evolution and the official statements of two major scientific institutions that exert a substantial influence on science educational policies in the United States. Given that science does have implications concerning the probable truth of supernatural worldviews, claims should not be excluded a priori from science education simply because they might be characterized as supernatural, paranormal, or religious. Rather, claims should be excluded from science education when the evidence does not support them, regardless of whether they are designated as ‘natural’ or ‘supernatural’.

Three more specific quotes from one or both papers:

- Claims about the supernatural should be prohibited from science classes not because they’re religious, but because science has “proven” them wrong.

- To exclude, a priori, the supernatural would validate the complaint voiced by some ID adherents and other creationists that science is dogmatically committed to Naturalism and thus opposed in principle to considering supernatural explanations (Johnson, 1999; see Stenger, 2006a). On the other hand, if there is no fundamental barrier preventing science from evaluating supernatural claims, then to declare the study of supernatural phenomena out of bounds to scientific investigation imposes artificial constraints on scientific inquiry, which potentially would deny science the noble task of purging false beliefs from the public sphere or the opportunity to discover aspects of reality that may have significant worldview implications.

- Naturalism is not a presupposition of doing science, but a conclusion from doing science.

And the second paper has a large section titled, Science Can Test Supernatural Claims: A Bayesian Perspective

They are both well worth reading.