Many large-scale applications of high-temperature superconductors depend crucially on the ability to achieve high critical-current densities J(c) (of the order of 10(5) A cm(-2)). Existing silver-sheathed (Bi,Pb)(2)Sr2Ca2Cu3Ox (BSCCO) tapes have J(c) values that come within about 25% of this target(1-4), these values being limited by the fact that the supercurrent flows percolatively around barriers that occur over many length scales(5). To elucidate the nature of these barriers, we shave measured the transport properties of individual filaments extracted from very-high-J(c) multifilament tapes(6). We find that J(c) for individual filaments reaches at least 8 x 10(4)Acm-2-about 50% higher than the average value over the whole cross-section of the wire. Although we injected the current to flow along the crystalographic a-b planes of the material, we found that all filaments possessed local characteristics of c-axis transport, indicating the presence of occasional nanometre-to micrometre-scale barriers at basal-plane-faced grain boundaries. An independent and much larger limiting influence on the critical current comes from unhealed cracks produced by deformation during the processing of the wires. These results provide direct evidence that better processing methods aimed at improving the c-axis alignment and at inhibiting residual cracks should raise the accessible J(c) values towards those needed for applications.