When I tried to use ～かねない to express my own feeling about a situation, I was
told by a native that it sounds incorrect and strange. However, ～かねる is
appropriate and fine. For example, this is the impression I get of what is
going on.

> 彼が本当のことを言っているとは信じかねる  
>  (the speaker thinks it's hard to believe he's telling the truth)
>
> 彼が本当のことを言っていると信じかねない  
>  (the speaker thinks he's probably not telling the truth, yet others think
> he is)

However, I was told that ～かねない cannot directly refer to the speaker's opinion,
but to can refer to someone else. For the second example, is it true that you
must directly reference a different party? Why is that?
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2012-08-03T07:19:37.177
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grammar usage
Why is it that ～かねる can refer to oneself, yet ～かねない must refer to another person?
Think of かねない as するおそれがある, and then think about how that would fit in with the
speaker's opinion. It would be like saying 信じるかもしれない when talking about
yourself. However, かねる simply means "cannot", so you can use it when speaking
about yourself.

I would translate your second sentence as:

> I might believe that he is telling the truth.
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2012-08-03T23:34:06.477
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I agree with @Jesse in that かねない is interpreted as おそれがある/しそうだ/ありうる/するかもしれない,
"Someone could do~~", "I fear someone might do~~", "Someone might do
~~(normally something unfavorable)".

I just suspect that your second sentence might be read as:

> (誰かが)本当のことを言っていると、彼が信じる恐れがある。

Where the subject for 言っている is someone except '彼', and the subject for 信じる is
'彼'.

For example, you might say it to mean:

> 彼が、(君が)本当のことを言っていると信じてしまうかもしれないよ。（だから、つまらない冗談はやめなよ。）  
>  "He might believe that you're telling him the truth / He might take you
> seriously (so you should stop telling him such a stupid joke.)"

Compare:

> 私にはわかり **かねます** 。  
>  _It's beyond me. / It's too hard for me to understand. / I'm afraid I don't
> know._  
>  彼なら嘘をつき **かねない** 。  
>  _I wouldn't put it past him to lie._
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