No, if a tank is properly cycled you will not see spikes in ammonia or nitrite. Did you cycle by ammonia dosing, an uncooked shirmp, or by the die off from your live rock? It sounds to me like your tank just ran out of ammonia source shortly before the end of your cycle. This would explain why small amounts of ammonia and slightly larger amounts of nitrite.
When you cycle a tank, you are growing the bacteria needed to consume ammonia and nitrite. These are 2 different types of bacteria (maybe even more). But for those that consume nitrite to grow, you first need the ones to consume ammonia. This is why you will see higher amounts of nitrite as the cycle is going on because they are "playing catch up" to the bacteria that consumes ammonia.
The only other thing that it could be is that there isn't enough area for bacteria to grow. You normally want to have 1-1.5 lbs of live rock per gallon. 40 is close, but how many lbs of sand do you have? If there isn't very much then that could be leading to the issue as well, meaning that you did fully cycle your tank...there just isn't enough area for the bacteria you need to support your livestock.
And final reason why you could have had a spike. How long did you spread out adding your fish? Adding too many fish at once can cause an ammonia spike. This with a tank that didn't have an ammonia source for too long could be the issue. Even a properly cycled tank that loses its ammonia source, by being fallow, will then have a spike in ammonia and nitrites due to the bacteria dying off from starvation.