I'm trying to reconcile two results on the classification of principal bundles. First, we have $\mathrm{Prin}_G(X)$ (the equivalence classes of $G$-bundles on $X$) is isomorphic to $H^1(X;G)$ (the first Cech cohomology group of X -- I'm taking $G$ to be abelian). Second, we have $\mathrm{Prin}_G(X)$ is isomorphic to $[X,BG]$, the set of homotopy equivalences of maps from $X$ to $BG$, the classifying space of $G$. If we now take $G=\mathbb{R}$, the real line viewed as an additive group, the first result seems to say we can have non-trivial bundles, while the latter seems to contradict that (since we can take $BG$ to be a point). How do I reconcile these? Does one result not apply in this case?

Thanks in advance!