Originally Posted by Bloomin Untidy Musician

My feeling is that your lack of playing experience is to blame and not the instrument/mouthpiece. Patience and disciplined practice provides the answer to most issues in playing. Yes sometimes a mouthpiece, or an instrument can be causing a problem, but at your early developmental stage you should be playing on a bog standard 7c size mouthpiece to give your muscles a chance to develop right across the register without using too much mouthpiece pressure. I find that beginner students nearly always find one of the starting harmonics (C or G) easy and one difficult. It sounds as though you might fall into this category.
By the way there is absolutely no problems with the bottom register of the cornet and low notes can sound fantastic. I can knock out pedale Bs and As with a trombone like sound. I do play on what might be termed as "a bucket" but i could still easily knock out a powerful pedale C on a smaller mpiece.
To answer your question, i think you are being a little impatient. If you are producing a nice sound from your instrument across other parts of the register, i would imagine that it is not wholly the instrument's fault. To be on the safe side, get another cornet player to have a blow on it just incase there are any holes! This could affect your sound significantly. Also check the water tap corks. If these are worn and the seal is broken this could be your problem.