Possible to do so, yes.
Higher than actually cleared ever set? NO. Never.
If you see higher than cleared on your display from the Mode-S output, then it was a error that I hope someone (you or me) is going to catch prior to the violation being issued!
B744/B748
ATC assigned altitude is set in MCP (mode control panel in Boeing terms). Knob is then pushed (IF level flight/already at a captured altitude) to initiate the climb to the higher level. This then inputs the new level to the FMS.
If I want a faster climb, I can then increase thrust (if using some type of reduced climb thrust was the default/previous setting), or reduce speed to trade some airspeed energy into climb rate if need to for a short time, or to reduce the speed towards best rate of climb speed.
Mode S reads from MCP, not FMS.
If I am cleared to "climb via the SID", then I set the highest altitude on the SID in the MCP and let the FMS do the work of doing the intermediate level off, if there are some. In this case, it ATC will not know from the Mode S that I am going to honor/comply with the level-offs, which is one drawback to the current mode S system.