If DH were negative, the process should be able to occur by itself. If DH were positive, the reaction could not occur by itself.
Indeed, almost all exothermic reactions are spontaneous at standard thermodynamic conditions (1 atmosphere pressure) and 25oC. However a number of common processes which are both endothermic and spontaneous are known. The most obvious are simple phase changes, like ice melting at room temperature. Also, many solids dissolve in water and simultaneously absorb heat.
Perhaps even more interesting, some processes which are not spontaneous at one temperature may be spontaneous at another--despite the fact that measurements show that DH changes only a little with temperature.
If we previously thought that there was a "natural" tendency to "favor" exothermic processes because strong bonds were formed, we now seem to have another competing tendency. But what is it?