We report on specific heat ($C_p$), transport, Hall probe and penetration depth measurements performed on Fe(Se$_{0.5}$Te$_{0.5}$) single crystals ($T_c sim 14$ K). The thermodynamic upper critical field $H_{c2}$ lines has been deduced from $C_p$ measurements up to 28 T for both $H|c$ and $H|ab$, and compared to the lines deduced from transport measurements (up to 55 T in pulsed magnetic fields). We show that this {it thermodynamic} $H_{c2}$ line presents a very strong downward curvature for $T rightarrow T_c$ which is not visible in transport measurements. This temperature dependence associated to an upward curvature of the field dependence of the Sommerfeld coefficient confirm that $H_{c2}$ is limited by paramagnetic effects. Surprisingly this paramagnetic limit is visible here up to $T/T_c sim 0.99$ (for $H|ab$) which is the consequence of a very small value of the coherence length $xi_c(0) sim 4 AA$ (and $xi_{ab}(0) sim 15 AA$), confirming the strong renormalisation of the effective mass (as compared to DMFT calculations) previously observed in ARPES measurements [Phys. Rev. Lett. 104, 097002 (2010)]. $H_{c1}$ measurements lead to $lambda_{ab}(0) = 430 pm 50$ nm and $lambda_c(0) = 1600 pm 200$ nm and the corresponding anisotropy is approximatively temperature independent ($sim 4$), being close to the anisotropy of $H_{c2}$ for $Trightarrow T_c$. The temperature dependence of both $lambda$ ($propto T^2$) and the electronic contribution to the specific heat confirm the non conventional coupling mechanism in this system.
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