It has been customary to accept that the observation of a highly deshielded proton is conclusive evidence that the molecule possesses a so-called low-barrier hydrogen bond (LBHB). To analyze this point, we have theoretically studied the features of the hydrogen bonds in hydrogen maleate and hydrogen malonate anions, both compounds experimentally characterized as LBHBs, and hydrogen oxalate anion, which has a hydrogen bond of the normal type. Ab initio electronic calculations along with a monodimensional approach to solve the corresponding nuclear Schrödinger equation are combined in order to obtain the ground vibrational energy levels and wave functions associated with the proton transfer in the three systems. According to our results, in the ground vibrational state the proton connecting the hydrogen bond has a maximum probability to be found in the region of the transition state for the hydrogen maleate and hydrogen malonate systems, so that they are LBHBs, whereas for the hydrogen oxalate the proton is localized near one of the two symmetrical minimum-energy structures. Combining the chemical shifts calculated at frozen structures with the probability density functions, we have calculated the 1H NMR chemical shifts for the three systems. Values of 22.85, 22.41, and 13.93 ppm for hydrogen maleate, hydrogen malonate, and hydrogen oxalate, respectively, have been obtained, results which are in good agreement with experimental values. Finally, these results allow us to discuss whether the appearance of a very high 1NMR chemical shift can be considered an unambiguous characterization of an LBHB. We postulate that an LBHB will always have an unusually downfield 1H NMR chemical shift, but the opposite statement is not necessarily true.