Hi Jenn,
It is best to move on.
Your boyfriend was not communicating with you wholeheartedly or consistently because the relationship is most likely just not working for him anymore and thereby he lacks the motivation to keep it going.
Yes, boredom could have indeed set in.
Even though it has been 2 weeks since you heard from him, it would be best to avoid contacting him again.
When he texted “Please give me more time” after you sent him a Happy Birthday message, he was indicating that he didn’t even want to share his special day with you in any way.
Following that Happy Birthday wish with apologizing and telling him about how much you missed him only made him feel that much more unwilling to talk to you.
He has reached a point where he no longer wants to meet or placate your emotional needs.
When you combine this sentiment with a strong sense of boredom and a predictability that he associates with his relationship with you, he is simply not motivated to try to work things out or save the relationship.
If you keep waiting to hear from him, you could keep yourself from moving on emotionally.
Unfortunately, he may have already moved on emotionally and may have only texted that you give him more time in the hopes that you would eventually just stop trying to contact him.