Diabetes-diabetes comorbidity
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