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Now, to answer your question: if you can't write a single method that satisfies both exception specifications, then you simply can't implement both interfaces in one class, end of story. This issue is specifically described in Gosling and Arnold's "The Java Programming Language."
I think he's talking about two interfaces with identical methods on them that differ only by their exceptions. That's perfectly legal, but their implementation must be the same method, and the exception signature of any implementation must be compatible with both superinterfaces.
In general, it's always OK to take exceptions out of method signatures for subtype relationships, but you can't add any in.