Here's the deal. "Cannot" is actually one word, unless one takes the effective, but inelegant route of splitting the two words with a modifier. As one word, spelled correctly, the not always modifies the can. Once it's split, it's all bets off.
The first two, I treated the 'not' as a modifier to the word directly preceding it, and 'can or can not' as 'can or can not get away with.' That's how I was taught.
In the third, the 'can not' directly modifies the verb following it, and implies 'can not perform this action,' to my personal intuitive connotation.
It also occurs to me that unclear modals get Americans into trouble quite often. We use "can" when we mean "may," as in "to have permission" or "to be allowed." Hence, the "get away with."
Trouble there, though, is that we also use "may" when we mean "might."
To me they mean precisely what they say they mean. The comment does not remotely infer as to whether or not the person's inability to do something has to do with that action not being allowed.
X = "do-something" in what I've added. Note that NOT do-something should not (necessarily) be construed as gold-bricking, as in the example of X = "smoke cigarettes" where the NOT do-something is a worthy accomplishment.
The first reads to me as: Expressing a frequent ability to NOT do-something. This to me implies that "NOT do-something" is the speaker's preferred way of handling this situation and it is a positive type of statement.
The other two, to me, read similar to each other, but rather differently than the first. These two are negative-type statements, expressing either rare ability or frequent inability.
The second: Expressing an infrequent ability to do-something.
The third: Expressing a frequent inability to do-something.
I think that broke my brain. I don't talk that way, except for the last case, "I often cannot X", by which I would mean: "I try and try, but I often cannot get the lid of the damned jar!". Or some such.
For the other cases, I think I am much more likely to use words like "usually", or "rarely".
Okay, here's my summary, in which I attempt to evaluate the three forms as succinctly as possible.
"I can not often X" indicates absence of opportunity.
"I often can not X" indicates absence of ability.
"I can often not X" is a non-traditional construction; it will generally only occur in spoken (i.e., non-formal) English, and in contrast to the other two constructions which indicate an absence, it indicates a positive ability to abstain from X.
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X = "clean toilets"
X = "masturbate"
Not that those two things are related in my mind, I just wanted two rather different examples to consider.
And you know, there's always the titillation factor.
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"I can often not-X" means that I can often do a thing with is the opposite of X.
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I suspect that I need a nap. ;)
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In the third, the 'can not' directly modifies the verb following it, and implies 'can not perform this action,' to my personal intuitive connotation.
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"Most of the time, I X with impunity, but frequently I am not able to X."
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Trouble there, though, is that we also use "may" when we mean "might."
Arse.
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I do not see how those three statments are not all equivilent.
So says the Dyslexic.
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The comment does not remotely infer as to whether or not the person's inability to do something has to do with that action not being allowed.
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The first reads to me as: Expressing a frequent ability to NOT do-something. This to me implies that "NOT do-something" is the speaker's preferred way of handling this situation and it is a positive type of statement.
The other two, to me, read similar to each other, but rather differently than the first. These two are negative-type statements, expressing either rare ability or frequent inability.
The second: Expressing an infrequent ability to do-something.
The third: Expressing a frequent inability to do-something.
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Here's how I'd read those sentences:
What does "I can often not X" mean to you?
--> I might try frequently, but it usually doesn't work.
What does "I can not often X" mean to you?
--> It is rare that I am able to... (as in, the opportunity is absent)
What does "I often can not X" mean to you?
--> Most of the time, I am not able to... (the ability is generally absent)
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For the other cases, I think I am much more likely to use words like "usually", or "rarely".
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concis
"I can not often X" indicates absence of opportunity.
"I often can not X" indicates absence of ability.
"I can often not X" is a non-traditional construction; it will generally only occur in spoken (i.e., non-formal) English, and in contrast to the other two constructions which indicate an absence, it indicates a positive ability to abstain from X.
Re: concis
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"I can not often X" and "I often can not X" mean the same thing to me. Most of the time I cannot or am not able to X for some unspecified reason.