Blowing up a singular variety Let $f:X\rightarrow Y$ be a proper surjective morphism. Suppose X have only normal crossing singularity. Let H and E are two irreducible components of the singular locus of X i.e, $Sing(X)=H\cup E$. So clearly H and E are codimension 1 sub varieties. Suppose $f|_{X\setminus E}$ is an isomorphism i.e f is isomorphism outside $E$. Also suppose $f|_E$ is a $\mathbb{P}^n$ bundle over $f(E)$. Is it possible that f is not a blow up along f(E)?
 A: Yes, it is possible. 
Well, it is kind of cheap, but if $f$ is not projective, then there is no chance of it being a blow-up, so you might as well say that $f$ is projective. Also, I assume you mean $f(E)$ with the reduced subscheme structure.
Let $Y$ be the quintessential example of a non-Cohen-Macaulay surface: Take a smooth surface and glue two of its points "transversally". By this I mean that $Y$ along its only singular point would be analytically locally isomorphic to $Y'=Z(x,y)\cup Z(z,t)\subseteq \mathbb A^4_{x,y,z,t}$.
For the following computation this is what matters, so I will assume that $Y=Y'$. Blow up the singular point of $Y$: the two components (branches, really) separate and you get a $(-1)$-curve on each. Now glue the two surfaces together along their just created $(-1)$-curves to form a normal crossing singularity. This glued together "double" $(-1)$-curve is your $E$ and the glued surfaces together is $X$ (if you started with an irreducible $Y$, then $X$ will also be irreducible). If you must have an $H$ as well, then blow up another point on $E$, but $H$ is really a red herring. 
Anyway, the original blow up morphism on each component gives a morphism 
$f:X\to Y$ which contracts $E\simeq \mathbb P^1$ to a single point, that is, to the singular point of $Y=Y'$. However, if you blow up that point, then the two components (branches) get separated, so $f$ is not the blow-up of $f(E)$.
If you really want a $\mathbb P^n$-bundle with $\dim f(E)>0$, then just take a product of all of this with your favorite smooth variety. If you want $n>1$, then do this with higher dimensional smooth varieties meeting in a single point. 

References for your questions in the comments below:


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*Gluing two surfaces along a curve should be pretty simple, but one could also just appeal to Karl Schwede's paper Gluing schemes and a scheme without closed points, especially Theorem 3.4 and Corollary 3.9. 

*See [Hartshorne, III.7.17].

