WAR DEPARTMENT, Washington City, December 5, 1863.
Mr. President: A general summary of the military operations of the past year is furnished by the report of the General-in-Chief, herewith submitted...
Exchanges under the cartel are now stopped, mainly for the following reasons:
First. At Vicksburg over 30,000 rebel prisoners fell into our hands, and over 5,000 more at Port Hudson. These prisoners were paroled and suffered to return to their homes until exchanged pursuant to the terms of the cartel. But the rebel agent, in violation of the cartel, declared the Vicksburg prisoners exchanged; and, without being exchanged, the Port Hudson prisoners he, without just cause, and in open violation of the cartel, declared released from their parole. These prisoners were returned to their ranks, and a portion of them were found fighting at Chattanooga and again captured. For this breach of faith, unexampled in civilized warfare, the only apology or excuse was that an equal number of prisoners had been captured by the enemy...
EDWIN M. STANTON,
Secretary of War.
I found Stanton's letter, explaining how the exchange cartel broke down following Vicksburg, in a PDF at the Villages Civil War Study Group website.
As far as Bredell's situation is concerned, I think we have enough evidence to state that he was captured (or surrendered) at Vicksburg on July 4, 1863, signed his parole form, stating that he would not take up arms against the United States until such time as he was exchanged, that same day and was most likely shipped to Mobile, Alabama, where he arrived no later than the middle of August 1863. The Confederates then unilaterally declared that he, and the other Vicksburg parolees were exchanged and he was, in the view of his government, free to rejoin the fight.