Is it possible to have both type-1 (insulin-dependent) and type-2 (non-insulin-dependent) diabetes mellitus? That is, to have both insulin resistance and zero (or negligible) insulin production? If not, why not? And if so, then how common is this (among humans with each type of diabetes)? Any information on why the comorbidity may be high or low is also most welcome.
Yes you could think of an individual as having both(1). A good place to start looking at "newer" diabetes other than the classic 1 and 2 is latent autoimmune diabetes (LADA). Though, I don't know if one would say you had no insulin production. It would seem as though patients exhibit some characteristics of both types(2), and indeed there maybe a continuum possible(3) between ether "pure" type on each extreme.
(1) The type 2 diabetes-associated variant in TCF7L2 is associated with latent autoimmune diabetes in adult Europeans and the gene effect is modified by obesity: a meta-analysis and an individual study. Diabetologia. 2012 Mar;55(3):689-93. doi: 10.1007/s00125-011-2378-z. Epub 2011 Nov 23.
(2) Latent autoimmune diabetes (LADA) is perched between type 1 and type 2: evidence from adults in one region of Spain. Diabetes/Metabolism Research and Reviews 2013; 1520-7560.
(3) Latent autoimmune diabetes in adults: evidences for diabetes spectrum?.Chin Med J 2013;126:783-788