  A space $X$ is "sequentially $n$-connected" at $x\in X$ if for every $0\leq
k\leq n$ and sequence of maps $f_1,f_2,f_3,\dots:S^k\to X$ that converges
toward a point $x\in X$, the maps $f_m$ contract by a sequence of
null-homotopies that converge toward $x$. We use this property, in conjunction
with the Whitney Covering Lemma, as a foundation for developing new methods for
characterizing higher homotopy groups of finite dimensional Peano continua.
Among many new computations, a culminating result of this paper is: if $Y$ is a
space obtained by attaching an infinite shrinking sequence $A_1,A_2,A_3,\dots$
of $(n-1)$-connected CW-complexes to a one-dimensional Peano continuum $X$
along a sequence of points in $X$, then there is an injection $\Phi:\pi_n(Y)\to
\prod_{j=1}^{\infty}\bigoplus_{\pi_1(X)}\pi_n(A_j)$ that is canonical after a
certain choice of paths in $X$ is made. Moreover, we characterize the image of
$\Phi$ using generalized covering space theory. As a case of particular
interest, this provides a characterization of $\pi_n(\mathbb{H}_1\vee
\mathbb{H}_n)$ where $\mathbb{H}_n$ denotes the $n$-dimensional Hawaiian
earring.
