Can a person convicted of a felony 25 years ago ever legally purchase a firearm and/or obtain a concealed carry permit?

Without additional action, the answer is generally no.  Once a person is considered a convicted felon, Texas law will not permit them to purchase a firearm for a number of years.  If it's been 25 years since the conviction date, however, Texas law does permit a felon to have a gun-- but only in their home.  But, keep in mind, that a person must comply with state and federal laws.  Federal firearms laws are similar, except they do not contain an exception for having a gun inn the home.  So.... even if a person could have a firearm under the state statute legally, they could potentially be prosecuted federally as a prohibited person in possesion of a firearm.  Bottom line, a 25 year convicted felon cannot purchase or carry under one or both statutes. 

The reason I included "without additional action" is that a conviction will continue to stand on someone's record until it is changed.  You cannot appeal a 25 year old conviction, but you may have some other appellate, post conviction remedies, like a writ, which could affect the conviction.  A couple of years ago, a slew of older cases got tossed because attorneys were not telling clients about the risks of deportation (Just as an example)  To know if you would be eligible for this type of post-conviction remedy, take your paperwork to an attorney that routinely handles criminal appeals.  A majority of criminal trial attorneys do not handle appeals--- so make sure you find one that knows and understands appeals--- it's a completely different beast.

Another possible avenue is a pardon by the governor.  In Texas, these are extremely hard to get because it's such a conservative state, but not impossible.  When you visit with the appellate attorney, mull over this option as well.  Sometimes a case/conviction will be wrong-- but not wrong enough for reversal with a writ.  In that case, sometimes governors will say, "okay, I'll make the record right."  Just don't hold your breath on this option.  However, if the need to carry a firearm is important, it's worth a shot.