how to append Attributed Text String with Attributed String in Swift

Use NSMutableAttributedString to achieve that.

Example

Swift 5

let yourAttributes = [NSForegroundColorAttributeName: UIColor.black, NSFontAttributeName: UIFont.systemFont(ofSize: 15)]
let yourOtherAttributes = [NSForegroundColorAttributeName: UIColor.red, NSFontAttributeName: UIFont.systemFont(ofSize: 25)]

let partOne = NSMutableAttributedString(string: "This is an example ", attributes: yourAttributes)
let partTwo = NSMutableAttributedString(string: "for the combination of Attributed String!", attributes: yourOtherAttributes)

partOne.append(partTwo) 

Swift 3

let yourAttributes = [NSForegroundColorAttributeName: UIColor.black, NSFontAttributeName: UIFont.systemFont(ofSize: 15)]
let yourOtherAttributes = [NSForegroundColorAttributeName: UIColor.red, NSFontAttributeName: UIFont.systemFont(ofSize: 25)]

let partOne = NSMutableAttributedString(string: "This is an example ", attributes: yourAttributes)
let partTwo = NSMutableAttributedString(string: "for the combination of Attributed String!", attributes: yourOtherAttributes)

let combination = NSMutableAttributedString()

combination.append(partOne)
combination.append(partTwo)

combination represents your final string which contains both formattings provided by yourAttributes and yourOtherAttributes

Even older

let yourAttributes = [NSForegroundColorAttributeName: UIColor.blackColor(), NSFontAttributeName: UIFont.systemFontOfSize(15)]
let yourOtherAttributes = [NSForegroundColorAttributeName: UIColor.redColor(), NSFontAttributeName: UIFont.systemFontOfSize(25)]

let partOne = NSMutableAttributedString(string: "This is an example ", attributes: yourAttributes)
let partTwo = NSMutableAttributedString(string: "for the combination of Attributed String!", attributes: yourOtherAttributes)

let combination = NSMutableAttributedString()

combination.appendAttributedString(partOne)
combination.appendAttributedString(partTwo) 

@glace's answer, modified to avoid empty NSMutableAttributedString declaration. Valid in Swift 3.1:

let yourAttributes = [NSForegroundColorAttributeName: UIColor.blackColor(), NSFontAttributeName: UIFont.systemFontOfSize(15)]
let yourOtherAttributes = [NSForegroundColorAttributeName: UIColor.redColor(), NSFontAttributeName: UIFont.systemFontOfSize(25)]

let partOne = NSMutableAttributedString(string: "This is an example ", attributes: yourAttributes)
let partTwo = NSMutableAttributedString(string: "for the combination of Attributed String!", attributes: yourOtherAttributes)

partOne.append(partTwo)

partOne is then your final string with all the attributes. No intermediate "combiner" necessary.

Swift 4

let yourAttributes: [NSAttributedString.Key: Any] = [.foregroundColor: UIColor.black, .font: UIFont.systemFont(ofSize: 15)]
let yourOtherAttributes: [NSAttributedString.Key: Any] = [.foregroundColor: UIColor.red, .font: UIFont.systemFont(ofSize: 25)]

let partOne = NSMutableAttributedString(string: "This is an example ", attributes: yourAttributes)
let partTwo = NSMutableAttributedString(string: "for the combination of Attributed String!", attributes: yourOtherAttributes)

partOne.append(partTwo)

Swift 5

As per "glace" answer, I just update font attribute and swift version.

    let boldFontAttributes = [NSAttributedString.Key.foregroundColor: UIColor.black, NSAttributedString.Key.font: UIFont.boldSystemFont(ofSize: 17)]
    let normalFontAttributes = [NSAttributedString.Key.foregroundColor: UIColor.darkGray, NSAttributedString.Key.font: UIFont.systemFont(ofSize: 15)]
    let partOne = NSMutableAttributedString(string: "This is an example ", attributes: boldFontAttributes)
    let partTwo = NSMutableAttributedString(string: "for the combination of Attributed String!", attributes: normalFontAttributes)

    let combination = NSMutableAttributedString()
    
    combination.append(partOne)
    combination.append(partTwo)
    lblUserName.attributedText = combination