There is certainly something wrong with your example
*Their budget has been unreasonably cut consistently, ...
but it's not clear what. Here is one possibility. "Unreasonably" and "consistently" are both the same type of adverb, specifically manner adverb,
Their budget has been cut in an unreasonable manner.
Their budget has been cut in a consistent manner.
and according to a theory of Zeno Vendler's, you can't have two adverbs of the same type in the same clause unless you conjoin them. Conjoining would give
Their budget has been cut unreasonably and consistently.
and that does sound better. However, the adverbs might not be manner adverbs. Maybe "unreasonably" is a sentence adverb, with the sense
It was unreasonable that their budget was cut.
But then, it's hard to find a reasonable interpretation for "consistently". Was it the consistency that was unreasonable?
??It was unreasonable that their budget was cut consistently.
That doesn't make sense. Lack of consistency would not make the cuts more reasonable.
The sense of "consistency" here is probably that cuts were made every single year, without exception, and I'm not sure this makes it a manner adverb.
It's easy to find ways of fixing the sentence, but it's more challenging to figure out what has gone wrong with the original version.