Statistical properties of the Kalman estimates under Gaussian noise
The following two statements are equivalent of saying:

$$
E(\hat{x}_{k|k} - x_k) = 0
$$

(1) That the estimator is unbiased; and 

$$
P_{k|k} = Var(\hat{x}_{k|k} - x_k)
$$

(2) That the estimator is consistent.

Both of these conditions are necessary in order for the filter to be optimal - i.e. the best possible estimate of $\mathbf{x}_{k|k}$ with respect to some criteria.

If (1) is not true, then the mean-square error (MSE) would be the bias plus the variance (in the scalar case). Clear, this is larger than the variance only and hence suboptimal.

If (2) is not true (i.e. the filter-calculated covariance is different to the true covariance) then the filter will also be suboptimal.  Since the Kalman Gain is based on the calculated state covariance, an error in the covariance will lead to an error in the gain.  Error in the gain means a suboptimal weighting of the measurements.

(As it happens, both conditions are true for a properly modelled filter. Errors in modelling, such as the dynamic model or noise covariances will also render the filter suboptimal).

Source: Bar-Shalom, especially Section 5.4 on page 232-233.