  Given a Tychonoff space $X$, we call a sequence $\langle\mu_n\colon
n\in\omega\rangle$ of signed Borel measures on $X$ a finitely supported
Josefson--Nissenzweig sequence (in short a JN-sequence) if: 1) for every
$n\in\omega$ the measure $\mu_n$ is a finite combination of one-point measures
and $\|\mu_n\|=1$, and 2) $\int_Xf\,\mathrm{d}\mu_n\to0$ for every continuous
function $f\in C(X)$. Our main result asserts that if a Tychonoff space $X$
admits a JN-sequence, then there exists a JN-sequence $\langle\mu_n\colon
n\in\omega\rangle$ such that: i)
$\mbox{supp}(\mu_n)\cap\mbox{supp}(\mu_k)=\emptyset$ for every $n\neq
k\in\omega$, and ii) the union $\bigcup_{n\in\omega}\mbox{supp}(\mu_n)$ is a
discrete subset of $X$. We also prove that if a Tychonoff space $X$ carries a
JN-sequence, then either there is a JN-sequence $\langle\mu_n\colon
n\in\omega\rangle$ on $X$ such that $|\mbox{supp}(\mu_n)|=2$ for every
$n\in\omega$, or for every JN-sequence $\langle\mu_n\colon n\in\omega\rangle$
on $X$ we have $\lim_{n\to\infty}|\mbox{supp}(\mu_n)|=\infty$.
