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Actions of $T^n$ on simply-connected $n+2$-manifolds were constructed in Theorem 4.7 of this paper. However, I haven't checked that the action is smoothable. The authors are only considering locally smooth actions in the paper (see p. 170 of Bredon for the definition of locally smooth); however, the construction yielding Theorem 4.7 is quite explicit, and seems to be at least piecewise-smooth. All such actions were subsequently classified by McGavran. So presumably one could use this classification to determine if there are smooth actions.