How do I get a list of all possible sums in a list nested list?

Total[Tuples@rn, {2}]

should do.

For cases where a very large number of tuples would be generated,

Fold[Total[Tuples[{##}], {2}] &, rn]

will generally be even quicker, and exhibit considerably lower memory pressure, to the point that it can be an order of magnitude faster because it can avoid paging if RAM gets low. If the ordering of the output is not important, sorting your $rn$ so that smaller lists are first (just a simple rn=Sort@rn) will increase the latter's performance even more.

Few additional alternatives:

Distribute[foo @@ rn, List, foo, List, Plus]

Flatten @ Outer[Plus, ## & @@ rn]

Activate @ Tuples[Inactive[Plus] @@ rn]