What is the point of introducing the concept of estimable function? I am reading about linear model, and it introduced the concept of estimable function. According to the book, a linear combination of the parameters $c^T \beta$ is estimable if and only if there exists a linear combination $a^T y$ such that $E(a^T y)=c^T \beta$. 
What is the point of knowing whether a linear combination of parameters can be estimated? Isn't it enough that we know all parameters in the model can be estimated?
 A: If you want to test $H_0: c^T\beta=0$ vs $H_1: c^T\beta\neq 0$, you will want to be able to estimate $c^T\beta$. 
Or, for example, if you need a prediction interval for $x_\text{new}\beta$, it would really help a lot if it's actually possible to estimate $x_\text{new}\beta$ ... (here we have $c=x_\text{new}^T$).
When your design is not of full rank it's useful to be able to distinguish what can and can't be estimated.
A: Let me give some perspective from linear algebra.
In linear model $ y = X\beta +\epsilon$, $E(a'y) = a'X\beta$ so the definition actually says that $c'\beta$ is estimable if and only if $c\in C(X')$ where $C(X')$ is the row space of $X$.
So if $X$ is full rank then any $c'\beta$ would be estimable.
But what if $X$ is not full rank? This definition is born to answer this question.
For any design matrix, full rank or not, we can estimate $c'\beta$ as long as $c$ is in the row space of $X$.
You can have a look at the pseudoinverse to find more intuition.
A: In the general situation, we have a model that is parametrised by $\theta \in \Theta$. We are interested in estimating $\theta$, or estimating some function $g$ of $\theta$.
We say that $g(\theta)$ is estimable if an unbiased estimator of $g(\theta)$ exists. That is, if there exists a statistic $T(Y)$ (a function from the data to $\mathbb R$) such that
$$ \mathbb E_{\theta} (T(Y)) = g(\theta) $$
for all $\theta \in \Theta$.
In the linear model scenario, our model is parametrised by $\beta \in \mathbb R^p$. Our data is the value of dependent variable $y$. We only consider linear functions. Therefore the definition is as follows:
$g(\beta) = c^T\beta$ is estimable if there exists a statistic $T(y) = a^T y$ such that
$$\mathbb E_{\beta} (a^T y) = c^T \beta,$$
for all $\beta \in \mathbb R^p$.
