When following the cycling method here there will be things happening that do not show up on the test kits. This is especially true for nitrite. In a basic cycle with no seeding, every ppm of ammonia one adds to the tank has a maximum level of both nirite and then nitrate that can be created. These nukbers are based on using an API type kit:
1 ppm of ammonia can become a max. of 2.56 ppm of nitrite which can become a max of 3.46 ppm of nitrate. However, some ammonia may be lost to evaporation which would reduce the numbers.
In cycling the danger are too much ammonia or nitrite being in the tank. For ammonia you do not want to exceed 6.4 ppm and for nitrite the level is 16.4 ppm. Both these conditions can stall of kill of a cycle if they are exceeded. The problem is that most nitrite kits stop at 5 to 8 ppm and during the cycle nitrite will get above these levels. But we cannot see how much on a test kit that stops at 5 ppm. It is possible to do diluted testing to know the actual levels of nitrite when the test shows they are at the max. level the kit can show. But this makes life harder for folks new to fish and doing their first cycle.
What was done in the creation of the cycling method here was to make doing diluted testing unnecessary. If one follows the directions for doing the cycle to the letter (I.e. does not add bigger or more frequent additions of ammonia), it is not possible to have either too much ammonia or nitrite in one's tank.
What you are not seeing is that nitrite is rising above 5 ppm, peaking and then coming back down. Once it hits 5 ppm and moves higher, all you know is 5 ppm untl it has come back down under 5 ppm.
The nitrite bacteria do not begin to reproduce until there is actually nitrite present. By this time one has gotten the ammonia ones to be multiplying and handling more and more ammonia and this creates the nitrite. So the A bacs get a good head start on the N bacs. Since the nitrite ones reproduce at a slower rate than the ammonia ones, nitrite is slower to get under control.
Aside from following the directions here for cycling, the one other component one needs to add to the cycling process is patience. If your dosing follows the directions, you will end up with a cycled tank. The average is about 5 weeks but some go faster and some slower. It depends on how much bacteria naturally comes into a tank at the start and. Unless we add some, we have no way to know how much we might have to start.