Is it in general true that $\nabla_\mu T^{\mu\nu}=0$ implies the matter equations of motion? I know of several cases where the covariant conservation of the energy momentum tensor $\nabla_\mu T^{\mu\nu}=0$ can be used to derive the equations of motion of the matter fields. Is this in general true? 
 A: Joshua N. Goldberg's "The Equations of Motion" (in Gravitation: An Introduction to Current Research, ed. Louis Witten, 1962) answered this question Yes $\nabla_{\mu}T^{\mu\nu}=0$ gives the equations of motion.
Some additional references:


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*Joshua N. Goldberg, "Strong Conservation Laws and Equations of Motion in Covariant Field Theories". Phys. Rev. 89 263 (1953)

*J.N. Goldberg, "Conservation Laws in General Relativity". Phys. Rev. 111 315 (1958)

*Joshua N. Goldberg and Peter Havas, "Lorentz-Invariant Equations of Motion of Point Masses in the General Theory of Relativity". Phys. Rev. 128 398 (1962)

A: Since $\nabla_\mu T^{\mu\nu} = 0$ has four free indices, it suffices to give equations of motion for matter with four degrees of freedom. The case of non-interacting particles ("dust" in general relativistic continuum fluid mechanics) falls into this category, and their geodesic motion follows from the conservation law.
However, you may need more degrees of freedom depending on what exactly you are evolving. For instance in a fluid with a velocity and two thermodynamic degrees of freedom you need another equation, often taken to be conservation of comoving particle number density $\nabla_\mu (nu^\mu) = 0$ or comoving rest mass density $\nabla_\mu (\rho u^\mu) = 0$. This is the classic case of compressible Euler flow. Imagine the difference between a stable fluid and one in which massive particles could annihilate into photons (while conserving energy and momentum) -- these would have very different behaviors.
Another example of needing another equation is ideal magnetohydrodynamics -- the above case but with the fluid being a perfect conductor, i.e. having no electric field in its rest frame. You would add something like $\nabla_\mu ({}^*F^{\mu\nu}) = 0$.
A: Informal, short answer would be: $\nabla_a T^{ab} = 0$ can be used to deduce at least part (but not necessarily whole) of the matter field equations. A recent discussion about this question can be found in sections 5 and 6 of the paper
I. Smolić: "On the various aspects of electromagnetic potentials in spacetimes with symmetries", Class. Quantum Grav. 31 (2014) 235002. DOI: 10.1088/0264-9381/31/23/235002. arXiv: 1404.1936 [gr-qc].
