Why do I not get a p-value from this ANOVA in R? Here's the data:
> tires <- data.frame(Wear  = c(17, 14, 12, 13, 14, 14, 12, 11,
                                13, 13, 10, 11, 13, 8, 9, 9),
                      Brand = rep(LETTERS[1:4], 4),
                      Car   = as.character(as.roman(rep(1:4, each = 4))))
> tires
   Wear Brand Car
1    17     A   I
2    14     B   I
3    12     C   I
4    13     D   I
5    14     A  II
6    14     B  II
7    12     C  II
8    11     D  II
9    13     A III
10   13     B III
11   10     C III
12   11     D III
13   13     A  IV
14    8     B  IV
15    9     C  IV
16    9     D  IV

Now I fit a two-way ANOVA with interaction:
two.way <- aov(Wear ~ Brand + Car + Brand:Car, data = tires)

Finally, no p-values: 
> summary(two.way)
            Df Sum Sq Mean Sq
Brand        3  30.69  10.229
Car          3  38.69  12.896
Brand:Car    9  11.56   1.285

A regular two-way ANOVA (i.e., Wear ~ Brand + Car) gives me p-values:
> summary(aov(Wear ~ Brand + Car, data = tires))
            Df Sum Sq Mean Sq F value  Pr(>F)   
Brand        3  30.69  10.229   7.962 0.00668 **
Car          3  38.69  12.896  10.038 0.00313 **
Residuals    9  11.56   1.285                   
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Is there a way to interpret this? The interaction plot shows me that there is definitely interaction between Brand and Car so I am hoping to incorporate this into my model.
 A: I am going to propose an alternative in case anyone comes with the same doubt.
As it was stated previously, in the design of your model we couldn´t consider a posible interaction $\gamma_{ij}$ between the factors brand and car:
$$
y_{ij}=\mu+ \alpha_i+\beta_j+ \gamma_{ij}+\epsilon_{ij}
$$
specially because we lack observations to make an stimation of the parameters, so the model is saturated. However, we could propose  a test with a degree of freedom as follows:
$$
y_{ij}=\mu+ \alpha_i+\beta_j+ \lambda(\hat\alpha_i\hat\beta_j)+\epsilon_{ij}
$$
where the interaction is simplified as the product of the parameters estimated without the interaction. So what we are going to do is add a regressor variable of known data to the model without interaction, and $\lambda$ it´s going to be it´s regression coefficient. This was actually proposed by Tukey a bunch of years ago as the one degree of freedom test for non-additivity (interaction).
A: Your model is saturated.  Any model will use at least 1 degree of freedom.  You have 2 factors with 4 levels each.  They both require 3 additional degrees of freedom.  The interaction consumes another 9 degrees of freedom.  Summing those 1 + 3 + 3 + 9 = 16, but you have only 16 data.  Thus, there are no degrees of freedom left with which to determine the residual variability, form standard errors, or test any hypotheses.  
