the problems can compound each other. let's take an easier example to work with: let's say that you had a lag of 10 ms and monitor response time of 5 ms. you'd think that because the lag is bigger than the response time, you won't have problems, but that's not the case. It would only be the case if you started playing EXACTLY at a time that is divisible by 5. For instance, you start playing at 5ms, and after 10ms, your monitor will have refreshed twice and your game would have caught up to the lag. And this will continue to happen synchronously. However, let's say you start playing at the 3ms mark of your monitor. Your monitor will refresh 2ms later at the 5ms mark while your game is still at the 2ms mark, so far so good. Another 5ms later, your monitor refreshes at the 10ms mark, while your game is at 7ms. With me so far? However, after another 3ms, your game refreshes.... and your need another 2ms for your monitor to catch up.
And that's why it's good to have the fastest refresh rate possible for gaming. In your case, even with 1ms, it can still be divided into smaller units, so while your monitor may have the lowest chance of frame misses in the market, it's still possible once in a blue moon.