Deconvolution of the sum of three gaussian distributions Consider the sum of three normal random variables: 
$
R_{i,j}=A_{i}+B_{j}+C_{i,j}\,
$
where 
$
A_{i}∼N(μ_{A},σ_{A})
$
,
$
B_{j}∼N(μ_{B},σ_{B})
$
and 
$
C_{i,j}∼N(μ_{C},σ_{C})
$
. Assuming $A$, $B$ and $C$ are iid (independent), the density of $R$ is still normal, with mean $μ_{A}+μ_{B}+μ_{C}$ and variance $σ_{A}+σ_{B}+σ_{C}$. 
Suppose I observe a list of realizations of $R$ (about 500 observations) as well as the $i$ and $j$ of each realization (so I know if two realizations of $R$ share the same $j$ for example).
How can I recover the distribution (mean and variance) of $A$, $B$ and $C$?
 A: You can estimate the three variances by fitting a mixed model with $R_{i,j}$ as the response and the factors $i$ and $j$ included as random effects.  All three variances would be identifiable but not only the sum of the $\mu$'s.
The following R code simulating data and fitting the mixed model
i <- factor(rep(1:20,each=25))
j <- factor(rep(1:20,25))
set.seed(1)
A <- rnorm(nlevels(i), mean=1, sd=1)
B <- rnorm(nlevels(j), mean=2, sd=2)
C <- rnorm(length(i), mean=3, sd=.5)
R <- A[i] + B[j] + C
library(lme4)
summary(lmer(R ~ (1|i) + (1|j)))

gives the estimates
Linear mixed model fit by REML ['lmerMod']
Formula: R ~ (1 | i) + (1 | j)

REML criterion at convergence: 944

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-2.9493 -0.6611 -0.0223  0.6345  3.5898 

Random effects:
 Groups   Name        Variance Std.Dev.
 i        (Intercept) 0.8373   0.9150  
 j        (Intercept) 3.0804   1.7551  
 Residual             0.2612   0.5111  
Number of obs: 500, groups:  i, 20; j, 20

Fixed effects:
            Estimate Std. Error t value
(Intercept)   6.1824     0.4432   13.95

that are pretty close to the true values.
A: Since $$\overbrace{\mathbf R}^{\text{vector of}\atop\text{$R_{ij}$'s}}=\overbrace{\mathbf X}^{\text{matrix of}\atop\text{$1$'s and $0$'s}}\underbrace{(\mathbf A^\top \mathbf B^\top)^\top}_{\text{vector of}\atop\text{$A_i$'s and $B_j$'s}} + \overbrace{\mathbf C}^{\text{vector of}\atop\text{$C_{ij}$'s}}$$the problem can be rewritten as a standard Normal regression problem with Normal priors on $\mathbf A$ and $\mathbf B$
