> Certyfikat tego banku jest ważny jeszcze prawie 2 lata do 10.07.2014.

Fakt + cert. obejmuje:
internetowy.pbsciechanow.pl
pbsciechanow.pl
wnioski.pbsciechanow.pl

> Masz po prostu niezaktualizowaną listę certyfikatów w przeglądarce.

I tak i nie, tzn. nowszych aktualizacji pod XP nie ma :).
FF jest najnowszy, IE w wersji 8 (bo to XP),
XP z najnowszymi latami + najnowszy rootsupd.exe.

Wina jest po stronie banku - brakuje im jakiegos
intermediate/chain certificate ichnim na serwerze.

To to:
The certificate not trusted error indicates that the SSL certificate is not
signed or approved by a company that the browser trusts. This occurs most
often for one of the following reasons:
- The web site is using a trusted SSL certificate but it is missing a
chain/intermediate certificate. Most trusted certificates require that you
install at least one other intermediate/chain certificate on the server to
link your certificate up to a trusted source.
[...]
Occasionally, certain browsers will give this error when others do not. For
example, Microsoft Internet Explorer can automatically download intermediate
certificates the first time you visit a site that needs one while Firefox
cannot. Once a trusted certificate is installed properly, all browsers will
work without getting this error.

Odpalilem ta strone w IE 8 i dziala,
no to odpalilem jeszcze raz w FF 14.0.1 i dziala (IE zaktualizowal baze).