I think the answer is no. The "t" in a t-test refers to the fact that $$\frac{N - m}{s}$$ is distributed according to the t distribution. So a t-test always assumes that there is some fixed mean $m$; it may be unknown, but it should be fixed. In your case, the null hypothesis encompasses many values for the mean of one population. So I don't think a standard t-test would apply.