A partial answer: At least in characteristic 0 and finite dimension this is impossible.
The trace of an idempotent matrix equals its rank (as can be seen by choosing a basis that extends a basis for the range of the linear operator). On the other hand, the sum of the four traces must be $\mathop{\rm Tr}I$, which is the dimension of the entire vector space. But since $E_1+E_2+E_3+E_4=I$, this means that the ranges of the four operators must be direct summands of the entire space. Then choose a basis for each of the summands, combine them to a basis for $V$ and write down the matrices in this basis. For each row only one of the matrices is allowed to have nonzero entries, but since the sum of the matrices is given, this determines them completely, and it is seen that indeed they partition the identity.
On the other hand, there's a trivial solution in characteristic $3$: simply let $E_1=E_2=E_3=E_4=I$.