The Bookings has already started and the user's role is Researcher, Student, or Guest

If the user has a Researcher, Student, or Guest role, they are not able to cancel a booking once it has started. Now, if the user has created a continuous booking, done by dragging the booking slot over multiple days/weeks, the system recognizes it as one booking and the user will not be able to cancel the booking. Please note, on the lab or my dashboard the booking might appear to be broken up into individual bookings, however it may still be one continuous booking depending on how it was created.
Additionally, if you go to Equipment>Bookings page, you can see what's the frequency of the booking (single or recurring). For recurring bookings, even though the user might not be able to cancel the current one, she will still be able to cancel any future bookings.

The user can edit/cancel bookings on individually assigned equipment but cannot do it on equipment assigned via a group

The user might have been given permission to edit/cancel bookings on their individual profile but not in any groups they are part of. If the user has access to Equipment X via Group 1, but Group 1 does not allow users to edit or cancel bookings, the user will not be able to edit/cancel any bookings on Equipment X, even though their individual allows them to do it on other pieces of equipment. This is because access to Equipment X has been assigned via Group 1, so the user is tied to the rules set in that Group. In this instance, either edit the booking for the user or change the Group rules. Please note, all users in that Group will be impacted by the change