There are a couple of different avenues.
With respect to disability benefits, an individual who may not be successful in applying for disability benefits the first time has access to a Veterans Affairs-funded lawyer through the Bureau of Pensions Advocates, who will support them in their appeal to the Veterans Review and Appeal Board.
Generally, the stats are that about 30% of people who may be declined for their application will go through the the Bureau of Pensions Advocates and go forward with a case to the Veterans Review and Appeal Board. Sometimes they're counselled not to: There's not enough evidence, it's been determined essentially that there may not be a service relationship and they may not be successful. Many others, as has been indicated here, go forward to the Veterans Review and Appeal Board and are given a hearing and may be found favourable in those cases.
There are other appeal processes for other Veterans Affairs programs that involve treatment benefits, rehabilitation, education and training benefits. If they're not happy with the decision that comes the first time, there are many appeals processes for veterans to pursue, to give to another audience and have another opportunity for a favourable decision.