I used to think this, until I inadvertently let a registration fail to renew because I didn't update my expired credit-card's billing details with NameCheap - they did send me automated emails about it but I missed them, unfortunately. Their grace-period is only 30 days and I didn't notice the problem until 45 days had passed when my domain-name was bought-up by a spam-site-scammer and redirected the site to a porn/virus-downloader site (yes, those still exist).
I paid $1500 (uugghhh) for the UDRP process to get the domain-name back ($1000 UDRP fee, $500 for the lawyer to do the paperwork), and the UDRP panel ruled against me: their response reasoning made it clear that they never actually looked at my submitted evidence - and unfortunately that $1000 is nonrefundable, gaaaaah. I still haven't gotten that domain-name back. (I will say that my previous other UDRP cases all ruled in my favour; I don't know why/how I somehow drew a crappy arbiter in this case, I'm just vexed that they can rule against me without any right to appeal; I expected better).
I use a prepaid provider. If at the day of the renewal there is no money "in the bank" they immediately release the domain. So yeah, I treat their reminders with priority. I really should change providers...