The problem with this is that we only have humans to go on for human-level intelligence. There are many highly intelligent creatures including but not limited to primates, crows, dolphins, and octopodes.
If intelligence on this level is what you're after, then there is your proof that an animal can achieve that level of intelligence. For higher intelligence, we're forced to look at what it is believed to be the reasons behind human intelligence.
There are several theories regarding the importance of intelligence in humans during evolution.
One theory being that in order to sleep in trees without falling out, we had to have higher brain functions. If our ancestors fell out of the trees, they could seriously injure themselves or get eaten by a predator. Bad things happened otherwise. Unicorns don't sleep in trees traditionally (though maybe yours do, but I'll assume that's not your case).
Another theory being that it was very much in the interests of survival to make friends and therefore be social. Higher brain functions are required to be able to recognize many faces and have many friends. If you had few friends which took care of you, eventually they may die off leaving you to fend for yourself, and being alone for the type of lifestyle which our ancestors had was a rough life to be sure. There was strength in numbers. Your unicorns may move in packs, but that's not quite the same as being social. They would need to be able to communicate in some fashion, perhaps using a highly sophisticated calling mechanism? Hmm, I'm not honestly convinced of this.
The traditionally believed reason behind modern human intelligence has been the usage of tools. In fact for a long time, it was believed the use of tools was a unique trait of human intelligence and what distinguished us from other animals. We now know this is no longer the case. Crows have been found to use sticks to obtain items out of reach, for instance. Although this isn't a unique trait, it could still be that the need for more advanced tools such as spears and knives meant the difference between being cold and hungry and being clothed and well-fed. Unicorns obviously cannot use tools, so I think we're also going to have to exclude this possibility.
Unicorns don't quite fit the description of these theories, and as such, none of the same conditions exist for a unicorn which existed for our ancestors. Though perhaps it is incorrect to base conclusions on the intelligence of human beings, however this is all we have to base ourselves with.
I hate to say it, but I'm in agreement with your sentiment that such animals could not have human level intelligence. It could be quite clever without being as intelligent as human beings, but it could never equal our intelligence without otherwise meeting some of these conditions in my humble opinion.