Will some rockets really collapse under their own weight?
The following references "rather definitively" confirm the claim.
The whole section, and indeed the whole document, is a "must read" for people who have got as far as reading this :-).
Chapter 7 - Materials for Launch Vehicle Structures
Grant Henson, Chief Scientist, Invariant Laboratories LLC, Westlake, Ohio
- . The next most common is the “steel balloon” design, which is very
thin-walled and not structurally stable under the load of its own weight
unless pressurized or stretched. Its stability before fill and pressurization
is maintained by pressurization with an inert gas or by mechanical tension
applied by a holding cradle. This design was most famously applied in
the Atlas missile.
- Balloon Tanks
The Atlas vehicle designed by K.E. Bossart at Convair Division of General
Dynamics in the early 1950s is exemplary of this type of design. The
other notable application is the Centaur upper stage, also developed
by General Dynamics. The Atlas maintained the balloon tank design
through several ICBM variants, the early Atlas E and F space launch
vehicles, and the Atlas I, II and III commercial space launchers. The
Centaur stage still uses the balloon tank design.
Balloon tanks require either mechanical tension (“stretch”) or internal pressure to keep them
from collapsing under their own weight prior to operation. In operation,
the pressure required for propellant feed is sufficient to keep the tank
stable under flight loads.
The following information is taken primarily from the review by Martin 
Everyone quotes this paper - when you find a copy please tell me :-)
- Richard E. Martin, "The Atlas and Centaur 'Steel Balloon' Tanks: A Legacy of Karel Bossart
Surprisingly good discussion here with substantial mention of Atlas and how the SpaceX practices have evolved from them
Why SpaceX Built A Stainless Steel Starship
Page 93 - Atlas pressurisation needed here
And page 100 ref 7 here
Welding of maraging steels 43 pages.
WARNING - If you already own too many books DO NOT OPEN THIS LINK !!! :-)
To Reach the High Frontier: A History of U.S. Launch Vehicles