Yes, immunity from prosecution can be granted, at which point a person in the US can't refuse to testify on the basis of self incrimination, because whatever they won't say can't be used to prosecute them - per the grant of immunity.
However, grants of immunity are typically specific, not a general absolution of any criminal behavior. The prosecution agrees not to charge the person on specific crimes, to get them to testify in court, presumably to go after a much greater criminal.
In this case, a grand jury wants Manning to testify. Unless the questions fall into the specific grant of immunity given to Manning, she can take the 5th.
Instead, Manning refuses to answer any questions. Her actions are now contempt of court, and typically for cases like this, she goes to jail until either she testifies, or the grand jury's term expires. Another grand jury can be empaneled to replace the expired one, can ask the same question, and can jail her again if she refuses to testify.
This is a telling line from that article:
“I will not participate in a secret process that I morally object to,
particularly one that has been used to entrap and persecute activists
for protected political speech,” she said in a statement released
after she was taken into custody.
Morally objecting to a secret process is not grounds for refusing to testify. It is also well established in US law that an unauthorized release of classified information is not protected political speech.