Yes, Slasher, I can, and here's why:
One of the conditions for Speed Trim Operation is '5 seconds following release of trim switches' (737 tech manual).
Therefore, if you have flown out-of-trim for five seconds - and that's a long time to be out of trim (try counting it out loud) - then Speed Trim will operate.
If you fly for five seconds or more with a requirement for trim, then you are not trimming properly.
I sometimes trim during the rotation, in response to an obviously wrong loadsheet trim for take-off.
Incidentally, if you often depart in the circumstances you describe, you should be a worried man/woman - you're clearly not making any money!!
