Transformation of E.coli cells
The two plasmids have to have different origins of replication. Two plasmids from the same incompatibility group cannot survive together in the same cell. 

This quote is from the Wikipedia entry on plasmids:


  Plasmids can also be classified into incompatibility group. A microbe can harbour different types of plasmids, however, different plasmids can only exist in a single bacterial cell if they are compatible. If two plasmids are not compatible, one or the other will be rapidly lost from the cell. Different plasmids may therefore be assigned to different incompatibility group depending on whether they can coexist together. Incompatible plasmids normally share the same replication or partition mechanisms.


This is a nice definition/explanation from an OpenAccess article:


  A formal scheme of plasmid classification is based on incompatibility (Inc) groups (Novick, 1987). The procedure for incompatibility grouping is based on the introduction, by conjugation or transformation, of a plasmid of an "unknown" Inc group into a strain carrying a plasmid of a known Inc group. If the resident plasmid is eliminated in the progeny, the incoming plasmid is assigned to its same Inc group (Datta and Hedges, 1971). Plasmids with the same replication control are "incompatible", whereas plasmids with different replication controls are "compatible". On this basis two plasmids belonging to the same Inc group cannot be propagated in the same cell line (Datta and Hughes, 1983; Couturier et al., 1988). Inc group identification has been frequently used to classify plasmids. The method has been an important tool to trace the diffusion of plasmids conferring antimicrobial resistance and also to follow the evolution and spread of emerging plasmids (Anderson et al., 1977).


A synopsis of the Datta &amp; Hedges article sited above:


  Classification
  In 1971, Hedges and Datta proposed a plasmid classification scheme based on the stability of plasmids during conjugation (incompatibility). The first incompatibility (Inc) groups were IncI, IncN, IncF, and IncP. Currently, 26 Inc groups are recognized in Enterobacteriaceae including six IncF (FII to VII) and three IncI (I1, Iγ, I2) variants. Major plasmid families occurring in Enterobacteriaceae are HI2, HI1, I1-γ, X, L/M, N, FIA, FIB, FIC, W, Y, P, A/C, T, K, B/O, FII, FIII, FIV. IncF plasmids are low- copy-number plasmids, often carrying more than one replicon with at least one replicon strongly conserved.


One way to think about it is: if one prepares a plasmid cloning vector to accept foreign inserts, and ligates in a "shotgun" collection of fragments from some DNA of interest. After transformation, selection and growth on plates, none of the colonies whose plasmids carry inserts ever contain more than one unique plasmid.  In other words, it may be possible for two different inserts to be inserted during ligation into the same vector backbone, but it is not possible to find a single pure colony that contains cells harbouring two different plasmids, each carrying a different insert. 

If a cell gets transformed by two different plasmids from the same inc group only one plasmid will "win."

Examples of compatible ori's commonly encountered:


pMB9 (pBR322, pUC, etc., etc.,)
F (fosmids)
bacteriophage lambda (some cosmids)
bacteriophage M13 and relatives (also yields ssDNA phage useful for Sanger sequencing)
bacteriophage P1
pSC101


Here is a nice table, but I cannot vouch for its accuracy.