I will be going to Ukraine soon from New York. Some of my family members in Ukraine were asking us to bring some phones. How many phones can I take to Ukraine if the other phones are packed into the suitcases and I take mine with me?
As personal belongings, which are exempt from duties and declaration, you can have up to TWO cell phones per person (article 370 there).
Declaration and formal duty and VAT payment in most cases make no sense as the same stuff will be cheaper (I suppose in most cases it should read as "much cheaper") in Ukraine. Having said that, I would not follow advice to "bring as many as you want and pay duties".
You will be most likely arriving at KBP and chances are high that your boxes in suitcases will draw no attention of customs officer. If you take chances to take 3rd and 4th with you, you will want to unpack them and somehow make them look as used rather than new. In other ports of entry customs officers are less loaded with work and some of them enjoy playing a game of picking tourists who are not aware of high import duties and trying to get them. If you are asked about how many presents you are having with you for your Ukrainian relatives, then know it's a question to estimate if you have anything that should be formally taxed (and the tourist is in turn a catch of the officer).
Also FYI items with declared value up to 150 EUR crossing border in mail are free from duties. And another thing is that some of the commercial carriers (related recent article on the topic) have informal ways to get around [high] standard duties for their own fee. I recently read they are not doing it anymore, but I'd double check with them - I used this kind of service a few times and it's normally the most cost efficient way to get US stuff here.