Yes you can. An appraisal is not absolute, and is not tied to your property as the only appraisal and appraised value available. That being said, you may find that in the current market your property value may have decreased and you may run into similar problems with other appraisals as well.

As far as your original question, even though the appraisal was done on your home, and you may have paid for it, the bank is the appraiser's client, not you. That means that the appraisal is the bank's property, and another bank would want to get their own appraisal from their own approved appraiser. If you were to go to a new lender or broker you would get a new appraisal done, although you may still have a value issue.

Ask your loan officer for a copy of the appraisal and tell them you want to challenge the value of the appraisal. If it was a weak appraisal that resulted in the lower value you can try to find better comparable sales that can be presented to the appraiser and the bank to substantiate a higher value. You should be able to do this by contacting another appraiser yourself or a Realtor. They should be able to research comparable sales to tell you if there are more similar sales than what are included on your initial appraisal. If you can, you may be able to get the original appraiser to reconsider the valuation, and have a chance at resurrecting your refinance.