In my limited experience, I have found certain carriers are more reliable than others when it comes to truthfully reporting capabilities and fulfilling duties such as remembering to report level or back on route (I am a non-radar controller, these things are extremely important to my ability to form and execute a good plan, and decreasing my workload). In turn, I am more likely to go out of my way to accommodate the carriers that help me out on a regular basis.
Me: XXX123 (at flight level 370) say when able flight level 390
XXX123: unable F390
Me: XXX123 descend and maintain flight level 350 due to traffic
XXX123: we can now accept flight level 390
Me: XXX123 flight level 390 no longer available, descend and maintain flight level 350
same carrier, obviously deviating (aircraft is ADS, thus automatically reporting its position without pilot action)
Me: XXX123 cleared to deviate 20 miles right of course
XXX123: unable
Me: !!! (looking for nearest baby to punch)
Now, all that being said, it is my job to provide good service to all aircraft and above all to be safe, but if I am busy and an aircraft from that carrier is stuck at F330 and is requesting F370, rerouting that aircraft to get a better altitude will be very low on the priorities list.