Can a model for non-negative data with clumping at zeros (Tweedie GLM, zero-inflated GLM, etc.) predict exact zeros? A Tweedie distribution can model skewed data with a point mass at zero when the parameter $p$ (exponent in the mean-variance relationship) is between 1 and 2. 
Similarly a zero-inflated (whether otherwise continuous or discrete) model may have a large number of zeros.
I'm having trouble understanding why it is that when I do prediction or calculate fitted values with these kinds of models, all of the predicted values are non-zero.
Can these models actually predict exact zeros?
For example
library(tweedie)
library(statmod)
# generate data
y <- rtweedie( 100, xi=1.3, mu=1, phi=1)  # xi=p
x <- y+rnorm( length(y), 0, 0.2)
# estimate p
out <- tweedie.profile( y~1, p.vec=seq(1.1, 1.9, length=9))
# fit glm
fit <- glm( y ~ x, family=tweedie(var.power=out$p.max, link.power=0))
# predict
pred <- predict.glm(fit, newdata=data.frame(x=x), type="response")

pred now does not contain any zeros. 
I thought the usefulness of models such as the Tweedie distribution comes from its ability to predict exact zeros and the continuous part.
I know that in my example the variable x is not very predictive.
 A: Note that the predicted value in a GLM is a mean.
For any distribution on non-negative values, to predict a mean of 0, its distribution would have to be entirely a spike at 0.
However, with a log-link, you're never going to fit a mean of exactly zero (since that would require $\eta$ to go to $-\infty$).
So your problem isn't a problem with the Tweedie, but far more general; you'd have exactly the same issue with the ordinary Poisson (whether zero-inflated or ordinary Poisson GLM) for example, or a binomial, a 0-1 inflated beta and indeed any other distribution on the non-negative real line.

I thought the usefulness of the Tweedie distribution comes from its ability to predict exact zeros and the continuous part.

Since predicting exact zeros isn't going to occur for any distribution over non-negative values with a log-link, your thinking on this must be mistaken.
One of its attractions is that it can model exact zeros in the data, not that the mean predictions will be 0. [Of course a fitted distribution with nonzero mean can still have a probability of being exactly zero, even though the mean must exceed 0. A suitable prediction interval could well include 0, for example.]
It matters not at all that the fitted distribution includes any substantial proportion of zeros - that doesn't make the fitted mean zero (except in the limit as you go to all zeros).
Note that if you change your link function to say an identity link, it doesn't really solve your problem -- the mean of a non-negative random variable that's not all-zeros will be positive.
A: This answer was merged from another thread asking about predictions zero-inflated regression model, but it also applies to the Tweedie GLM model.
Regression-like models predict mean of some distribution (normal for linear regression, Bernoulli for logistic regression, Poisson for Poisson regression etc.). In the case of zero-inflated regression you predict mean of the zero inflated-something distribution (e.g. Poisson, binomial). When the probability density function of the non-inflated distribution is $f$, then probability density function of zero-inflated distribution is a mixture of point mass at zero and $f$:
$$ 
f_\text{zeroinfl}(y) = \pi \,I_{\{0\}}(y) + (1-\pi)\, f(y)
$$
where $I$ is an indicator function. Zero-inflated regression model predicts mean of $f_\text{zeroinfl}(y)$, i.e.
$$
\mu_i = \pi \cdot 0 + (1-\pi)\, g^{-1}(x_i'\beta)
$$
where $g^{-1}$ is an inverse of the link function. So since you are predicting the mean of this distribution, you won't see the excess zeros in your predictions since the zeros are not the mean of the distribution (while they shrink the mean towards zero), the same as linear regression does not predict the residuals. 
This is illustrated on the plot below, where values of random variable $Y$ are plotted against $X$, where $Y$ follows a zero-inflated Poisson distribution with mean conditional on $X$. The black points are the actual data that were used to fit the zero-inflated Poisson regression model, the red points are the predictions, and the blue points are means of $Y$ within the six arbitrary groups of $X$ values. As you can see, clearly the zero inflated Poisson regression model estimates $E(Y|X)$.

