question regarding COBRA notification (labor law)

hey!

my friends and I were discussing the fact that (most) employers are required to inform individuals of COBRA following a qualifying event, and that failure to do so can result in up to a 110$ fine for each day the notification was not given.

this got me thinking... if an employee experiences a qualifying event and is not informed of COBRA, but they are aware that they're supposed to be informed of COBRA, couldn't they just wait months and months before they inform their employer that they weren't notified of COBRA eligibility and get an increased 'award' (it's my understanding that the former employee is awarded the up-to-110$-a-day fine, and a quick purview of case history supports that notion).

what benefit does an employee who experiences a qualifying event have in immediately notifying their employer that they need to inform them of COBRA eligibility?

thanks for your time!