TOR connection to a VPN OK, VPN connection to TOR go to jail.
This assessment, in general, is not accurate. Though it's true that there's always something that can go wrong with any security system. Let's consider some user who uses one of these setups.
The Invisible Wanderer
The case where the speaker's assessment is most true, is when the user is physically traveling around to locations which are not tied to them.
In this case, the user is hoping they're physically indistinguishable by local surveillance systems, and to achieve k-anonymity among all of the other people who could have passed through those locations to access those network entry-points.
Connecting to a VPN with user log-in credentials now is clearly a problem. This is because, the user just went through all the trouble to be indistinguishable in that location, to then connect themselves, and their VPN account, to that location. This isn't so much a problem if the VPN provider also cannot identify the user. Which may be the case when the VPN account payments are not tied to the user and that account is only used once, for instance.
Although, how the user connects to the network is also important. Because, a user that vanishes to some unlinkable location, which then utilizes their phone, or personal hotspot, to access the network, also just identified themselves at that location.
But in general, the situation isn't made better by connecting to a VPN after Tor. If the user's VPN account points to their personal identity, it usually doesn't matter much if their physical location in some narrow time-window is unknown. Most users can be found physically once their personal identity is uncovered.
There isn't much essential difference between an ISP and VPN provider. They both sit between the user and the web, proxying everything (Imagine this illustration, but with a VPN provider added). The main difference is that the ISP's business model is selling internet access and selling or sharing user data; while the VPN business model is building user trust of their effectiveness in hiding user internet activity and location data (though it's not clear whether a VPN can be trusted)(0).
Most users will not have access to the kinds of resources necessary to do this well. The Invisible Wanderer may consider not using a VPN at all, or paying anonymously for throwaway accounts.
The Visible Wanderer
This user only uses public internet access-points. So, they're likely identifiable, on camera sitting at a café or library somewhere.
In this case, it doesn't hurt if the VPN account is accessed before touching the Tor relays, because the user's identity is already traveling to these locations. It's a safer bet to trust a VPN with knowledge of the Entry Guard than to trust an ISP with the same.
On the other hand, it does hurt if their VPN account is accessed after the Tor relays. Because again, a direct connection can be made going backwards from the thing the user is touching on the web, to their VPN account.
Many users can act as a Visible Wanderer. Here, it's better to connect to a VPN before the Tor relays.
The Stationary
This user is sitting in some private location that is linkable to their personal identity. This scenario is almost exactly like the Visible Wanderer case, except that the user is probably not on camera connecting every one of their keystrokes to some activity on the web.
In this case as well, for mostly the same reasons, it's a safer bet to connect to the VPN prior to connecting to Tor.
TL;DR
In all cases, the user has to limit activity fingerprinting, and increase the ephemerality and personal unlinkability to their accounts and devices. That'll help protect their anonymity.
While, in general, connecting to a VPN after Tor does not improve their anonymity. It's usually safer to do VPN-then-Tor, even though in some cases, using a VPN at all (as with any linkable account) may harm their anonymity.
