"I believe Bob's story."
Indicates a state of being. There is no real action still being performed. That is why "I am believing" sounds unnatural:
** "I am not believing Bob's story."
Although you could say this, it is unusual because it implies that you are still in the process of deciding whether or not you believe Bob's story. If this is the case, it would probably be better to use a different word:
"I am unconvinced by Bob's story."
Or use believe with not yet:
"I do not yet believe Bob's story."
I am (verb)+ing
When a verb is an action, it is quite natural to use it with the progressive form (-ing):
"They recite Shakespeare."
Recite is an action, so it is also quite natural to say:
"They are reciting Shakespeare."
Believe, Know, Think
Besides believe, another word which is usually only used to signal a state is know:
"He knows the answer to this riddle."
Know is just a verb, but here (and in most cases) it indicates the state of being that you know something. Yesterday, you didn't know Alice. Today, you meet Alice. Now you know Alice. For the same reason, saying "am knowing" or "is knowing" in this sense is unnatural:
** "He is knowing the answer to this riddle."
If you are still in the process of thinking of an answer, 'think' would be more appropriate:
"I am thinking of the answer to this riddle."
Note that 'think' can also be used to mean 'believe', and if you use it in that way, you have the same problem as with 'believe':
"I think Bob is telling the truth."
Here, think shows your state of mind. Thinking Bob is telling the truth is the same as believing Bob. Your thinking process has concluded and no further action is being performed. Thus, this sounds unnatural:
** "I am thinking Bob is right."
Of knowing, of believing
When speaking abstractly about words, "knowing", "believing" and so on can be used. But these also feel strange if you try to translate them to actions:
"The state of knowing everything is unobtainable. The more you know, the more you know what you don't know."
→ ** "No man is knowing everything."
→ "No man knows everthing."
"The state of being blissfully unaware of the plights of others."
→ ** "You are being blissfully unaware."
→ "You are blissfully unaware.
→ "You are ignoring the plights of others."
Ignore is a clear action, so using it in the progressive is quite natural.
"Know", "believe" and so on can be used commands (imperatives):
"To pass the test, know how to solve this problem without using a
"Believe me! I didn't mean to hurt him."
You could interpret these statements as literally meaning something like "please put yourself in a state of mind that you know how to solve this problem" or "please put yourself in the state of being that you believe me."