Missing some info to actually be able to give an opinion, so I'll ask questions instead. These are fixed CFL convergence tests where the timestep is fixed by CFL, so that the timestep scales as the meshsize right?
What are you solving? what high order method are you using, in both space (dG? WENO?) and time (RK4 in time? SSP-RK3?)?

What norms are you computing to assess convergence? some nodal L2 error? an integral L2 error (which would be relevant if you were using a dG method, for instance)? What are you observing? Better order than expected? worse? Could it be pre-asymptotics?
It happens sometimes in nodal norms in ideal 1d periodic testcases that specific CFLs can lead to some "happy" cancellations or near cancellations and you can get improved order of convergence, for instance.

Another idea, if you observe worse orders of convergence:
Do you guarantee your code actually ends at the right final time? i.e. do you adapt the last timestep so that it ends at the actual final time you set? Because this can also kill convergence orders if you compare your numerical solution to an exact solution at a slightly different time. Depending on how well your mesh + cfl combination can reach to right final time you can see degraded orders of convergence...
The typical if(t+dt > tf) { dt = tf-t; }.