Two professors declined to write LOR for me because they are busy? I had my M.A. degree in a public university in the States. After working in media for two years in China, I want to pursue research and teaching career so I started to plan for PhD in media psychology (under communication). 

I had my GRE tested (the last one has expired) in Sept and got good scores. But due to my heavy workloads and procrastination, I hesitated to contact two professors who instructed my thesis and know me relatively well until I finished SOP in November. After a week, with my follow-up letter, both of them refused to write me strong LOR.

My thesis advisor said "as the Director of PhD studies I am unable to write letters this year."And he suggested me to ask the other instructor. 

But the other one replied: "It is good to hear from you and nice to hear that you are applying to doctoral programs. Part of the reason that I was so delayed in my response is that I am a bit overwhelmed with work right now.  Given all of my existing demands,  unfortunately I do not think I am in a position to give the time and effort to write you a strong recommendation. I am sorry about this."I discussed about possibility of pursuing PhD with him when I took his seminar, but he was not very encouraging.

But reading between the lines (especially "part of the reason"), I feel like they are not confident that I will succeed as a researcher. I asked my thesis advisor for more feedbacks like whether he thought I am less than qualified and where to improve and if I should take another M.A in Psychology. But he did not reply.  ：（

I know to a large degree it's my ill plan to blame. But I am not certain if they are making excuses and whether to ask them for LOR next year. Is it often for faculty to refuse LOR because they don't see your potential with other excuses?
There's a chance they're trying to avoid writing the LOR because they don't think they can write one that's good enough to help you.

If an instructor can't write an LOR they think will actually help a student based on what they know to be the truth (even if only because they don't think they know you well enough), most instructors will try to encourage the student to go to someone else.  No one likes writing a bad reference.  But they also struggle with how to do that without hurting the student's feelings.  Some just won't tell you the bad news directly and they may foot-drag and hope you go away or they may make silly excuses about being busy or whatever.  That's spineless and it's not right, but that doesn't mean it doesn't happen.

So, if you ask someone for an LOR and they don't seem to want to do it, you should probably just ask someone else.