A recurring theme in economics blogs, particularly those that tend to be disparaging of
mainstream Keynesian theory, is that
Keynesians like to be New Keynesian (NK) when talking about theory, but Old
Keynesian (OK) when talking about policy. John Cochrane has recently made a similar observation, which is picked up by Megan McArdle. To take just one
example of this alleged sin, in the basic New Keynesian theory Ricardian
Equivalence holds (see below), so a tax financed stimulus should be as
effective as a debt financed stimulus, yet Keynesians always seem to prefer debt
The difference between Old and New that Cochrane focuses on relates to models of consumption. In the first year textbook OK model, consumption just depends on current income. The coefficient on current income is something like 0.7, which gives rise to a significant multiplier: give these consumers more to spend, and the additional spending will itself generate more output, which leads to yet more income, and so the impact of any stimulus gets multiplied up.
Basic NK models employ the construct of the (possibly infinitely lived) intertemporal consumer. To explain, these consumers look at the present value of their expected lifetime income, and the income of their descendents if they care about them (hence infinitely lived). This has two implications. First, temporary shocks to current income will have very little impact on NK consumption (it is a drop in the ocean of lifetime income). The marginal propensity to consume out of that temporary income (mpc) is near zero, so no multiplier on that account. Second, a tax cut today means tax increases tomorrow, leaving the present value of lifetime post-tax income unchanged, so NK consumers just save a tax cut (Ricardian Equivalence), whereas OK consumers spend most of it. However NK consumers are sensitive to the real interest rate, so if higher output today leads to higher inflation but the nominal interest rate remains unchanged, then you get a multiplier of sorts because NK consumers react to lower real interest rates by spending more.
So far, so different. But the NK consumption model assumes that agents can borrow whatever they need to borrow. There are good theoretical reasons why that is unlikely to be true (e.g. asymmetric information), and even better empirical evidence that it is not. Empirical studies that look for ‘natural experiments’, where agents obtain an unexpected increase in post-tax income which is likely to be temporary, typically find a mpc of around a third (even for non-durables), rather than almost zero as the basic intertemporal model would predict. (For just one recent example: Consumer Spending and the Economic Stimulus Payments of 2008, by Parker, Souleles, Johnson, and McClelland, American Economic Review 2013, 103(6): 2530–2553.)
mainstream Keynesian theory wants a more realistic
model of consumption, it often uses the (admittedly crude) device of assuming
the economy contains two types of consumer: the unconstrained intertemporal
type and the credit constrained type. A credit constrained consumer that
receives additional income could consume all of that additional income, so
their mpc out of current income is one.  That credit constrained consumer is
therefore rather Old Keynesian in character. But there are also plenty of
unconstrained consumers around (e.g. savers) who are able to behave like
intertemporal maximisers, so by including both types of consumer in one model
you get a hybrid OK/NK economy.
So it is perfectly possible to be an Old Keynesian and a New Keynesian at the same time, using this hybrid model. It may not be a particularly elegant model, and the microfoundations can be a bit rough, but plenty of papers have been published along these lines. It is a lot more realistic than either the simple NK or OK alternatives. It explains why you might favour a bond financed stimulus over the tax financed alternative, because there are plenty of credit constrained consumers around who are the opposite of Ricardian.
You can make the same point about one of the other key differences between OK and NK: the Phillips curve. The New Keynesian Phillips curve relates inflation to expected inflation next period, and assumes rational expectations, while a more traditional Phillips curve combined with adaptive expectations relates current inflation to past inflation. While I do not think you will find many economists using the OK Phillips curve on its own nowadays, you will find many (including this lot) using a hybrid that combines the two. The theoretical reasons for doing so are not that clear, but there is plenty of evidence that seems to support this hybrid structure. So once again it makes sense to be both OK and NK when giving policy advice.
Neither story is as exciting as the idea that New Keynesians are really closet Old Keynesians, who only pay lip service to New Keynesian theory to gain academic respectability. Instead it’s a story of how
Keynesian economists try to adapt their models to be more consistent with the
real world. How dull, boring and inelegant is that!
 I say ‘could’ here, because if the increase in income lasts for less time than the expected credit constraint, then smoothing still applies, and the mpc will be less than one.
 My own view is that the mpc out of temporary income is also significant because of precautionary savings: see the paper by Carroll described here.