what are chances I can get the same accomodations as the tour if I arrive a night early

Most of the time I have been able to get reservations at the tour hotel when I get the hotel list (not sure about London and Paris because I haven't done these). However, I have found that the rate quoted by the hotel when mentioning that I am on the RS tour often is higher than the rate I can find online.

that is quite a popular decision by many tour participants.

We usually provide info on your first and last night's hotel (or city-tour hotel) earlier than the rest of the list, just for this purpose. But you still don't have to worry about it for a while.

I usually book the extra nights as soon as I receive the tour hotel information. Ending the Adriatic tour in Dubrovnik in May, I was happy I had done it that early as several folks on the tour had to move to a different hotel for the extra night(s). Didn't seem to be a problem for any of us at the beginning of the tour, though. The hosts have always been most accommodating in assigning me the same room for the tour as for the extra nights both before and after.

We've always tried to arrive a day early. On our first tour, we didn't book quickly enough and had to come up with another hotel for 1 night. It was great, but we had to pack up. We stayed an addional night at the last hotel in Rome and had to move rooms. Last tour, we booked as soon as we got the hotel list. They knew we were on the tour and we had our room for the duration.

I am doing the 21 day BOE next week. I got the names of the beginning and ending hotels right away but...couldn't make up my mind as to what I thought I might want to do ahead of time. By the time I made up my mind the hotel website was showing full. I did a search on hotels in Haarlem and got onto Booking. Com. That site showed a vacancy at the aforementioned tour hotel so I emailed the hotel and they were able to book me directly.

My other thought is are you sure one night ahead is enough? Lol! Really, for your first stop I would do 2 nights ahead.

" By the time I made up my mind the hotel website was showing full."

Don't take this as the final word. E-mail the hotel directly; they may still be able to accommodate you. The Pension Peters in Berlin (a favorite of Rick Steves) says this on their website, and I can vouch that lots of hotels in Switzerland should too:

"If our system should answer, that there are no more rooms during the time you wish to visit us, please give us a call or drop an e-mail. Maybe we can make the impossible possible."

Indeed, when I went to Berlin, e-mailing got us a room here when the online system had no availability for the rooms we wanted. And for my upcoming trip to Switzerland, many hotels had websites only showing doubles, triples and quads, but e-mailing got me the single I needed, without difficulty, in multiple places.