There's nothing out of the box that does this in the API's (from the API point of view, it's still getting a tile and isn't trying to analyze the image).
It might be possible to use the TileLoaded event on the layer to check the image stream coming back and see if you can "recognize" that this is one of these images - ie first look at the stream length which is 2521bytes (although subject to change), and then check some bytes to see if these match the "not available" image bytes". However the only option you get is to replace the image with your own, or set it to "null", meaning not displaying an image at all (so it's not the blurred image you see in ArcMap).