Ask Mr. Smarty Plants

Every plant that puts down roots in the earth compete in some way with its neighbors; some compete more aggressively than others.  The rhizosphere, that is, the soil where roots grow, can be thought of quite accurately (if not a bit simplistically) as a war zone.  Plants, animals, fungi and microorganisms living within the rhizosphere continually struggle with one another for their share of the resources available there.

Many trees and other plants interfere with their competition through one of a multitude of processes known collectively as allelopathy.  Allelopathy typically involves living or dead and decaying plant parts exuding, volatilizing or leaching chemical compounds that are inhibitory or harmful to nearby plants.

Hackberry trees, Celtis spp. are known to inhibit the growth of competing plants through allelopathic processes.  The mechanism used by hackberries involves the release of a witch's brew of chemicals known as phenolic phytotoxins leaching from fallen, decaying leaves.  The leachates from rotting hackberry leaves have been shown to inhibit the germination of seeds and the development of seedlings.  So yes, hackberry can do harm to nearby plants, though it is unclear if it has any effect at all on established plants such as trees, shrubs or even mature perennials. 

Ashe juniper, which is probably the cedar your mentioned in your question, and oaks are also known to be allelopathic.  Few plant species will thrive beneath the canopy of Ashe juniper.  If there are any sycamores, pecans or walnuts in your garden, they're also waging chemical warfare on one another.  Lest you come to think of these trees as "bad plants," you should know that as time goes on we will certainly come to learn that many, if not all, plants employ chemical defenses (and offenses) in one form or another.

As landscape plants, hackberries are not without other problems.  They're messier than most other trees, tend to be short-lived, are prone to breaking in storms and don't often make particularly handsome specimens.  On the other hand, hackberries are extremely important sources of food for wildlife.  Many birds, mammals and some butterflies depend on hackberries for their survival.

On balance. hackberries definitely play an important role and have a place in nature and maybe even in peripheral areas of your landscape.  They are probably best left out of more prominent parts of your garden, though.