This paper presents robustness properties of the Kalman filter and the associated LQG/LTR method for linear time-invariant systems having delays in both the state and the output. A circle condition relating to the return difference matrix associated with the Kalman filter is derived. Using this circle condition, it is shown that the Kalman filter guarantees (1/2, infinity) gain margin and +/-60 degrees phase margin, which are the same as those for non-delay systems. However, it is shown that, even for minimum phase plants, the LQG/LTR method cannot recover the target loop transfer function. Instead, an upper bound on the recovery error is obtained by using an upper bound of the solution of the Kalman filter Riccati equations. Finally, some dual properties between output-delayed systems and input-delayed systems are exploited.