This is a very interesting conversation. The more I think about it, more I think Nathan is right. The claim of direct link between current account and (distinct set of) financial flows is widepread and wrong, even if there are indirect links mediated by exrates or other variable.
The claim you see very often is that for a change in the current account to take place, there has to be a distinct offsetting transaction on financial account - someone else has to make a separate investment decision to finance current account move toward deficit.
The claim is that there is an accounting-enforced link between two distinct balances. E.g. that the choice of Chinese units to increase their holdings of US assets must increase US current account deficit. Not just is one factor tending that way, but necessarily results in it.
This is definitely the impression you get from a lot of international finance textbooks, where the current account must balance with a financial account that cosists of portfolio flows and FDI, which are in turn aggregates of investment decisions made by wealth owners.
Nathan's point is bank positions are also component of financial account, and these are not actively chosen but are passive counterpart of transactions by other units. When a deposit is transferred between units from different countries, that shows up in financial account of BoP.
When I make a payment to someone in another country - whether for a good or an asset - the payment itself shows up as an offsetting entry in the financial account. There is no need for anyone else to do anything to maintain payments balance.
Suppose I’m in Canada and buy a good from the US. "Buy" means I transfer a deposit at either US or Canadian bank to seller. In 1st case, Canada’s foreign assets decline, in 2nd, Canada’s foreign liabilities increase. Either way, no further financial account transaction is needed.
So transaction itself creates offsetting BoP entry. As far as accounting goes, nothing more needs to happen. No one else's decision to make a decision to buy a Canadian asset or sell an American one.
Now, my purchase of US asset/good has created a long position in Canadian dollars, or short position in US dollars, somewhere in financial system. (In this perspective, that’s a better way of putting it than loss of foreign exchange.) That position may cause change in behavior.
Maybe added to Canadian banks' dollar liabilities. This may make them banks less willing to lend, pushing up interest rate. Or perhaps a US entity ends up with Canadian dollar deposit they don’t want, and their effort to find a buyer pushes down the value of the Canadian dollar.
These adjustments may lead to changes in cross-border flows that offset initial change. E.g., if Canadian banks end up with more dollar liabilities and this reduces their willingness to lend, that will slow income growth in Canada, which will mean less imports.
But the *possibility* of these kinds of offsetting behavioral changes is very different from the idea of a direct, one-for-one link between current account and foreign investment. You can't make any causal claims based on accounting identity between current and financial account.
Because of course many other factors influence exchange rate, interest rate, credit conditions, growth, and these can change without any corresponding changes in either current-account or foreign-investment flows.
The textbook model that suggests there is some unique point in interest rate-exchange rate space that balances independent trade and financial flows is deeply misleading, imo, and Nathan's post is helpful in seeing why.
There's a deep analogy here with the Keynesian liquidity-preference story about interest rate. Pre-Keynesians saw it as equating two distinct flows of saving and investment. Keynes showed that I always creates its own S, no need for interest rate adjustment to maintain equality.
The interest rate rather is determined as *stock* equailibrium, based on investors desire for more liquid assets. Similarly people are imagining exrate in terms of cross-border flows, but really it's a stock equilibrium based on portfolio preferences, not cross-border payments.
Should add that it's not a question of what it true in absolute sense but which perspective is more useful. I think conventional view is probably ok for small, less developed countries, but Nathan's mine is better for discussing transactions between large, rich countries.
In light of earlier discussion of whether passive accomodation by banking system breaks link between current account and cross-border investment flows (@NathanTankus and I say yes, @Brad_Setser and @DanielaGabor say no), I decided to look at cross-border bank deposits.
As Nathan points out in post that started discussion, cross-border deposit holdings are financial flows that are created as the flipside of any other cross-border transaction. Because payment is made by transferring deposits, all transactions are self-financing in very short run.
The debate is (in part) about whether this very short run is economically relevant. If deposit positions are always quickly unwound and there are no fx market makers who will take position for any significant time, then yes, net c.a. flows must equal desired change in portfolios.
(Even then, I wouldn't concede, since desired portfolios may be edogenous. If e.g. market particiapnts strongly expect exrate to be near E1=$1.15, then anything that pushes rate away from this will lead to speculative flows in other direction, offsetting any initial disturbance.)
But if in fact there are large accomodating positions passively accumulated in banking system and/or by fx specialists, then that clearly breaks direct link between cross-border portfolio choices and current account balances.
This is clearly the case in the euro system, where TARGET2 balances are purely accomodating. (This might have been a better example for Nathan's original post, actually.) But arguably same thing exists to lesser but still significant extent elsewhere.
So looking at the US, we see that for the last decade, foreign holdings of US bank deposits have increased faster than US holdings of foreign deposits by about 1% of US GDP/year. (These are flows, not change in stocks, so shouldn't be affected by valuation changes.)
In accounting terms, about half the US current account deficit over the past decade has been financed by changes in bank deposit holdings. 
It doesn't seem likely that much if any of these bank deposit holdings are the result of active portfolio choices - presumably for anyone looking to take a cross-border investment position, short-term government bonds would dominate bank deposits.
Seems more plausible that they are simply the counterparts of other transactions, which for whetever reason have not been unwound. Which suggests that it's at last plausible that large part of cross-border flows are passively accomodated and don't produce pressure anywhere else.
Another question that hasn't been raised is whether exchange rate movements will lead to trade adjustments even in principle. There's a lot of empirical evidence that in many cases the Marshall-Lerner condition is not satisfied - i.e. a depreciation can move trade toward deficit.
If that's the case - and at short horizons it certainly is, and probably at longer ones for many countries - then if a financial inflow leads to appreciation, it will actually move the trade balance toward surplus - the opposite of the accounting-based argument of Pettis.
