> Many VPN services claim that they do not maintain connection logs
If you're paying for the service, then you need to authenticate to it either with a password or certificate. Either way, the VPN provider knows when you're connected. Also, to prevent sharing of accounts, they will often limit the number of IP addresses you can connect from (within a certain period of time), which means that almost all of them log IP addresses and connection times. That information, combined with other evidence, could be enough to connect you to certain activities.
For example, suppose you're harassing someone on a forum and they get the cops involved. They know the date and time of all of your posts, so they go to your VPN provider, and you're the only person who was connected every single time a post was made. Now, it's good to catch people like that, but if you require strong anonymity just use Tor.
There is absolutely no reason they would have to to log ip addresses to prevent account sharing. They can easily limit you to one logon at a time or simply ignore sharing and limit each account's bandwidth, preferably rounded to the megabyte or coarser.
If you're paying for the service, then you need to authenticate to it either with a password or certificate. Either way, the VPN provider knows when you're connected. Also, to prevent sharing of accounts, they will often limit the number of IP addresses you can connect from (within a certain period of time), which means that almost all of them log IP addresses and connection times. That information, combined with other evidence, could be enough to connect you to certain activities.
For example, suppose you're harassing someone on a forum and they get the cops involved. They know the date and time of all of your posts, so they go to your VPN provider, and you're the only person who was connected every single time a post was made. Now, it's good to catch people like that, but if you require strong anonymity just use Tor.