We study the learnability of sums of independent integer random variables\ngiven a bound on the size of the union of their supports. For $\\mathcal{A}\n\\subset \\mathbf{Z}_{+}$, a sum of independent random variables with collective\nsupport $\\mathcal{A}$} (called an $\\mathcal{A}$-sum in this paper) is a\ndistribution $\\mathbf{S} = \\mathbf{X}_1 + \\cdots + \\mathbf{X}_N$ where the\n$\\mathbf{X}_i$'s are mutually independent (but not necessarily identically\ndistributed) integer random variables with $\\cup_i \\mathsf{supp}(\\mathbf{X}_i)\n\\subseteq \\mathcal{A}.$ We give two main algorithmic results for learning such\ndistributions:\n 1. For the case $| \\mathcal{A} | = 3$, we give an algorithm for learning\n$\\mathcal{A}$-sums to accuracy $\\epsilon$ that uses $\\mathsf{poly}(1/\\epsilon)$\nsamples and runs in time $\\mathsf{poly}(1/\\epsilon)$, independent of $N$ and of\nthe elements of $\\mathcal{A}$.\n 2. For an arbitrary constant $k \\geq 4$, if $\\mathcal{A} = \\{ a_1,...,a_k\\}$\nwith $0 \\leq a_1 < ... < a_k$, we give an algorithm that uses\n$\\mathsf{poly}(1/\\epsilon) \\cdot \\log \\log a_k$ samples (independent of $N$)\nand runs in time $\\mathsf{poly}(1/\\epsilon, \\log a_k).$\n We prove an essentially matching lower bound: if $|\\mathcal{A}| = 4$, then\nany algorithm must use $\\Omega(\\log \\log a_4) $ samples even for learning to\nconstant accuracy. We also give similar-in-spirit (but quantitatively very\ndifferent) algorithmic results, and essentially matching lower bounds, for the\ncase in which $\\mathcal{A}$ is not known to the learner.\n