Missing Hypothesis in Electromagnetism Texts In the Feynman Lectures, Chapter 21, I find the statement 
We have solved Maxwell's equations.  Given the currents and charges in any circumstance, we can find the potentials directly from these integrals and then differentiate and get the fields.
In Purcell's book  on electricty and magnetism, I find the statement
 Except for the possible addition of a constant field pervading all of space, the conditions $curl({\bf B}) =4\pi{\bf J}/c$ and $div({\bf B})=0$, uniquely determine the magnetic field of a given distribution of currents.
I don't have Griffiths's textbook in front of me at the moment but I'm pretty sure he says something similar.
Clearly all of these statements are false.  For example, if the current and charge distributions are identically zero, then I can solve Maxwell's equations by setting ${\bf E}=grad(f)$ and ${\bf B}=grad(g)$ where $f$ and $g$ are arbitrary harmonic functions, so that in particular ${\bf E}$ and ${\bf B}$ are by no means unique (even up to the addition of a constant vector field).
Presumably, then,  there is some hypothesis that Feynman, Purcell and others have omitted, possibly because they thought it was too obvious to mention.  What is that hypothesis?
 A: The assumption missing in all these statements is that there are boundary conditions assumed to be given.
E.g. for Poisson's equation $\Delta f = \rho$, the solution is unique for Dirichlet and/or Neumann boundary conditions, see e.g. section 1.9 in Jackson's "Classical Electrodynamics".
A: I think that the texts you quote are referring to localized charge and current densities and the fields are defined in the whole space. The natural requirement is that far from the sources the vector fields decay as $1/r^2$ or faster, uniformly in all directions. This is the hypothesis of an induction field which is appropriate for static fields. With this requirement the elactrostatic and magnetostatic equations uniquely determine a solution. Dealing with potential fields the requirement is that they decay as $1/r$ or faster. Your counterexample does not works, as your harmonic functions would be bounded (from below or from above) and thus they must  be constant in view of Liouville's theorem for harmonic functions in $R^n$. As they vanish at infinity they must be zero everywhere.
