100% anonymity is probably impossible.
My explanation of how the anonymity works - which is probably very incomplete and vague - is the following: you put some technical - and non-technical - efforts into making it more difficult and more costly to track your actions online. There are many aspects here (for example if you search Google while logged in, you can be tracked by Google no matter which VPN you use), but since you only mentioned VPN I assume you mainly concern about being tracked by ISP or someone accessing your traffic.
VPN increases the difficulty to track you for this group. However this difficulty only translates into more effort for the adversary in order to track you. This is definitely not an "ultimate protection", just the matter of whether someone really needs to uncover you.
And whether it is the case, this typically comes down to how you use this anonymity. And thus how willing the state (i.e. police) would be to spend their limited resources on you. Are you running an international criminal drug syndicate? Good chance. Are you illegally downloading some Hollywood movie? Not likely.
If the police want to trace you, they go to your ISP and ask what IP's I've connected too.
Note that here you have already lost your anonymity at this point, so you are probably now talking about evidence.
They then go to the VPN provider and ask for my activity, what if they don't log anything? Where do the police end up going after that?
They can order/force/coerce/convince the VPN provider (depending on country) to spy on you. This could be a very minor effort operation. This would strip down the VPN level anonymity completely. Unless you stop using this provider right away, if you continue your usage, they will have the full record of it.
Finally in real life there is no need even for that. Read about the busts on people who purchase guns on dark markets. An undercover police officer takes over a seller account, sells a gun, which is delivered to one's house (apparently many purchases simply use their residential address for that). After the delivery the police searches the premises, finds a gun, arrests the buyer. In this case the police not only doesn't need to go to VPN provider, they don't even need to monitor the buyer's traffic.