You can use the UITextViewDelegate shouldChangeTextInRange: method to limit the text entry to the height of the text view:
func textView(textView: UITextView, shouldChangeTextInRange range: NSRange, replacementText text: String) -> Bool {
    // Combine the new text with the old
    let combinedText = (textView.text as NSString).stringByReplacingCharactersInRange(range, withString: text)
    // Create attributed version of the text
    let attributedText = NSMutableAttributedString(string: combinedText)
    attributedText.addAttribute(NSFontAttributeName, value: textView.font, range: NSMakeRange(0, attributedText.length))
    // Get the padding of the text container
    let padding = textView.textContainer.lineFragmentPadding
    // Create a bounding rect size by subtracting the padding
    // from both sides and allowing for unlimited length 
    let boundingSize = CGSizeMake(textView.frame.size.width - padding * 2, CGFloat.max)
    // Get the bounding rect of the attributed text in the
    // given frame
    let boundingRect = attributedText.boundingRectWithSize(boundingSize, options: NSStringDrawingOptions.UsesLineFragmentOrigin, context: nil)
    // Compare the boundingRect plus the top and bottom padding
    // to the text view height; if the new bounding height would be
    // less than or equal to the text view height, append the text
    if (boundingRect.size.height + padding * 2 <= textView.frame.size.height){
        return true
    }
    else {
        return false
    }
}