Varying Order of Covariant Differentiation Does the order of covariant differentiation matter?
Will   E$_i$$_j$,$_k$$_l$$_t$ =E$_i$$_j$,$_l$$_t$$_k$ ?
Does it matter if the tensor E$_i$$_j$ is continuously differentiable?

In general no, for example on a reimannian manifold where the christoffel symbols and its derivatives do not vanish there is a loss in commutativity:
$$\nabla_a \nabla_bT^c \ne \nabla_b \nabla_aT^c$$
infact:
$$\nabla_a \nabla_bT^c - \nabla_b \nabla_aT^c = R^{c}_{dab}T^d$$
Where $R$ is called the reimann christoffel tensor. It has a beautiful story which you can find by searching for it in wikipedia along with a more geometric picture as a preamble to the concept of curvature.
In a euclidean space  this is not the case as the mixed partial derivatives are equal and the components of $R$ vanish. 
