It was first conjectured by Antieau, Gepner, and Heller (and is now a theorem due to Neeman) that for a noetherian finite dimensional scheme X, the category of perfect complexes over X has a bounded t-structure iff X is regular. I will talk about a general result that I proved jointly with Hongxing Chen, Kabeer M Rahul, C Parker, and Junhua Zheng (arxiv:2401.00130) that gives a new obstruction to the existence of bounded t-structures on triangulated categories satisfying a small finiteness condition. Our result, when applied to Perf(X) for X noetherian finite-dimensional, outputs the earlier result. Several other relevant existing results get generalized or improved in the process. I will discuss applications of our results to triangulated categories arising from E1-rings.