To expand my brief comment, I can point to specific examples involving finite groups of Lie type in the defining characteristic $p>0$.     The general set-up for reduction mod $p$ is somewhat complicated and involves a $p$-modular system in the Curtis-Reiner sense (a suitable valuation ring $R$ such as a ring of $p$-adic integers, with fraction field of characteristic 0 and finite residue field of characteristic $p$).  But an example such as $G = \mathrm{SL}_2(\mathbb{F}_p)$ suffices.    Here the irreducible representations and their characters over $\mathbb{C}$ have been known since the time of Frobenius; an elementary exposition is given <a href="https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Humphreys.pdf">*here*</a>.   Take $p$ "large enough" to avoid minor problems.  

For this finite group $G$, approximately half (= reciprocal of order of Weyl group) of the irreducibles belong to the "principal series" (induced from 1-dimensional characters of a Borel subgroup).     In turn, reduction mod $p$ of such a representation typically (though not always) involves two composition factors of different dimensions.    Here you get a concrete instance of the phenomenon you ask about.  A natural choice of $RG$-lattice for a typical principal series representation leads to an indecomposable module over the finite field (here $\mathbb{F}_p$) having two composition factors: these correspond to adjacent vertices in the Brauer tree of the block in question.     But these irreducible modular representations lift naturally to $RG$, where the direct sum of the two (tensored with the fraction field of $R$) yields the same Brauer character.

But in most cases for this $G$, having the same Brauer character implies having the same ordinary character.  (This is easily seen by comparing the known values.)   So this is a way of constructing distinct lattices which lead to decmposable and to indecomposable modular representations.         

[I mention this particular group $G$ since it was already studied by Brauer and his student Nesbitt in the late 1930s.   He must have realized that different lattices can give different modules for $G$, which probably stimulated his invention of "modular characters": traces of $p$-regular elements in $G$ when computed in $\mathbb{C}$ using lifted roots of unity.    These were later dubbed "Brauer characters" by Curtis-Reiner, since their values are not actually "modular".   Back in the 1980s I asked some experts such as Walter Feit and Jon Alperin about the question you raise, which unfortunately seems not to be discussed in the literature.  I should have asked Brauer himself in the early 1970s when I got acquainted with him, but it didn't occur to me then.]