Rh-mediated carbene (co)polymerisation of diazomethane works best in the presence of Rh(III) catalyst precursors, the use of which leads to a significant increase in polymer yield and molecular weight. Chain termination via β-hydride elimination is severely suppressed for these species, although this process does still occur leading to unsaturated chain ends. Subsequent chain walking leading to the formation of branched polymers seems not to occur. Computational studies describing pathways for both chain propagation and chain termination using a (cycloocta-2,5-dien-1-yl)Rh(III)(alkyl) species as a representative model for the active species revealed that chain propagation is favoured for these species, although β-hydride elimination is still viable at the applied reaction temperatures. The computational studies are in excellent agreement with all experimental results, and further reveal that chain propagation via carbene insertion (leading to linear poly-methylene) occurs with a much lower energy barrier than insertion of 1-alkenes into either the Rh-H bond after β-hydride elimination or into the Rh-C bond of the growing polymer chain (leading to branched polymers). These energetic differences conveniently explain why experimentally the formation of branches is not observed in (co)polymerisation reactions employing diazomethane. The formation of substantial amounts of low-M(w) oligomers and dimers in the experimental reactions can be ascribed to the presence of (1,5-cyclooctadiene)Rh(I) species in the reaction mixture, for which chain termination via β-hydride elimination is clearly favoured over chain propagation. These two species stem from a non-selective catalyst activation process during which the catalyst precursors are in situ activated towards carbene polymerisation, and as such the results in this paper might contribute to further improvements of this reaction.