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We investigate a number of problems related to infinite runs of weighted timed automata, subject to lower-bound constraints on the accumulated weight. Closing an open problem from an earlier paper, we show that the existence of an infinite lower-bound-constrained run is -- for us somewhat unexpectedly -- undecidable for weighted timed automata with four or more clocks. This undecidability result assumes a fixed and known initial credit. We show that the related problem of existence of an initial credit for which there exists a feasible run is decidable in PSPACE. We also investigate the variant of these problems where only bounded-duration runs are considered, showing that this restriction makes our original problem decidable in NEXPTIME. Finally, we prove that the universal versions of all those problems(i.e, checking that all the considered runs satisfy the lower-bound constraint) are decidable in PSPACE.