Echoes are known as modifications of the usual quasinormal ringing of a black hole at late times because of the deviation of space-time from the initial black hole geometry in a small region near its event horizon. We consider a class of brane-world model solutions of the Shiromizu-Maeda-Sasaki equations, which describe both black holes and wormholes and interpolate between them via a continuous parameter. In this way the brane-world scenario provides a natural model for wormholes mimicking the black hole behavior if the continuous parameter is chosen near the threshold with a black hole solution. We show that in the vicinity of this threshold interpolating between black holes and wormholes, quasinormal ringing of the wormholes at the initial stage is indistinguishable from that of the black holes with nearby values of the above parameter, but at later times the signal is modified by intensive echoes. We notice that the black hole mimickers that are wormholes near the threshold have the largest quality factor and are therefore the best oscillators among the considered examples.