When using \textsc, LaTeX issues warning: Font shape `OT1/cmr/bx/sc' undefined

You can try

\documentclass[a4paper]{amsart}
\usepackage[latin1]{inputenc}   
\usepackage[T1]{fontenc}   
\usepackage[finnish]{babel}
\begin{document}
\begin{center}
{\large \textbf{\textsc{ANSIOLUETTELO}}}

\textbf{\textsc{Ansioluettelo}}
\end{center}
\end{document} 

You did nothing wrong. TeX tells you that it does not have Small Caps bold Computer Modern font, so it switches to bold normal font. This is a design decision by font author (Knuth) not to have this variant; you may try to find fonts that have bold small caps (not that I know of such) or to fake the effect using uppercase.

Actually your line is already in uppercase, so I wonder whether you really need small caps: usually people use small caps with lowercase letters, with \textsc{\MakeLowercase{...}}

The libertine font provides bold small caps. Abstaining from capital letters in the argument of \textsc doesn't hurt, either. ;-)

\documentclass{article}

\usepackage[T1]{fontenc}
\usepackage{libertine}

\begin{document}

\Large

\textbf{\textsc{Ansioluettelo}}% right

\textbf{\textsc{ANSIOLUETTELO}}% wrong

\end{document}