Are 1 week Sprints really possible?
Before we try to answer the question of whether 1 week Sprints are possible, lets examine 'why' we would even want to consider that.
You could have a situation where a team is doing 2 week Sprints and they're looking to increase their velocity and a 1 week Sprint would add another level of urgency.
You could also have a situation where a team feels that Scrum (and Sprints) are not a good fit due to the constant fluctuation in their work. This is common in operation teams where they are often required to drop everything they're working on in order to deal with a top priority issue.
These are all valid reasons.
In the first situation, the team needs to realize that converting from a 2 week Sprint to a 1 week Sprint is not trivial. In fact, teams need to be really good at a 2 week Sprint before making the switch. So their level of automation (amongst other things) needs to be exemplary. If not, the 1 week Sprint will quickly highlight areas that need to be addressed. This may actually be a good approach for teams that are struggling to find ways to improve.
In the second situation, oftentimes the temptation is to switch to Kanban. Kanban allows for more fluidity and doesn't force the team to commit to unrealistic goals. Before switching to Kanban, the team may want to consider the 1 week Sprint approach. The shortened cycle may be more aligned with their work.
Not all teams are able to get to a 1 week Sprint and that isn't necessarily a reflection on the team. There could be too many organizational impediments that make a 1 week Sprint unreachable. But like anything in life, you'll never know unless you try.