Suppose that two compact topological manifolds with boundary have homeomorphic interiors. Can we conclude that the two manifolds are homeomorphic? What happens in the smooth category?
Gjergji Zaimi's answer gives a strong positive conclusion: the product of the boundaries with $\mathbb{R}$ are necessarily homeomorphic. I just want to add a couple of explicit examples illustrating that the boundaries themselves need not be homeomorphic.
The first example is given by Milnor in his article "Two complexes which are homeomorphic but combinatorially distinct", partially stated there as theorem 3. Consider the 3dimensional lens spaces $L_1=L(7,1)$ and $L_2=L(7,2)$. In that article, Milnor proves (section 2) that for any $n>1$, the interior of $L_1\times D^{2n+1}$ is diffeomorphic to the interior of $L_2\times D^{2n+1}$. However, Milnor also proves (at the end of section 4) that their boundaries, $L_1\times S^{2n}$ and $L_2\times S^{2n}$, have different Reidemeister torsions. Consequently, the boundaries cannot be homeomorphic, by the topological invariance of Reidemeister torsion.
To obtain other examples, we can apply the result stated by Oscar RandallWilliams' in his comment to Gjergji Zaimi's answer. It suffices to find $h$cobordant closed manifolds $M$, $N$ of dimension greater than 3 which are not homeomorphic: then $M\times I$, $N\times I$ will have diffeomorphic interiors by Oscar's argument, but their boundaries are not homeomorphic. As Milnor proves in the article cited above, we can take $M=L_1\times S^{2n}$ and $N=L_2\times S^{2n}$ for $n>0$. I wrote a more detailed discussion concerning examples of this type in an answer to a related mathoverflow question.

$\begingroup$ Thank you Ricardo, this is a very interesting answer, I didn't know this result of Milnor. $\endgroup$ Mar 31 '13 at 7:14

$\begingroup$ @Daniele: You are most welcome. I am glad I could help. $\endgroup$ Mar 31 '13 at 10:25
In "Uniqueness of the Open Cone Neighborhood" by K.W. Kwun, Proc. Amer. Math. Soc. 15 (1964) pp.476479, it is shown that if two manifolds have homeomorphic interiors then their boundaries have homeomorphic suspensions, or in other words the boundaries are hcobordant. I don't know if one can conclude anything more in this generality.

8$\begingroup$ I don't think one can say any more generally: If $W : M \leadsto N$ is an $h$cobordism then $S^1 \times W$ is an $s$cobordism (crossing with $S^1$ kills Whitehead torsion), and taking covering space corresponding to the subgroup $\pi_1(W) \leq \pi_1(S^1 \times W)$ shows that $\mathbb{R} \times M \cong \mathbb{R} \times N$. In particular, $[0,1] \times M$ and $[0,1] \times N$ have isomorphic interiors. $\endgroup$ Apr 25 '12 at 13:36

$\begingroup$ @Oscar. The conclusion that the two boundaries times $R$ are homeomorphic is the theorem of Kwun paper cited by Gjergji. $\endgroup$ Apr 25 '12 at 14:49