In a secret sharing scheme a dealer shares a secret <i>s</i> among <i>n</i> parties such that an adversary corrupting up to <i>t</i> parties does not learn <i>s</i>, while any <i>t</i>+1 parties can efficiently recover <i>s</i>. Over a long period of time all parties may be corrupted thus violating the threshold, which is accounted for in <i>Proactive Secret Sharing (PSS)</i>. PSS schemes periodically rerandomize (refresh) the shares of the secret and invalidate old ones. PSS retains confidentiality even when <i>all parties</i> are corrupted over the lifetime of the secret, but no more than <i>t</i> during a certain window of time, called the refresh period. Existing PSS schemes only guarantee secrecy in the presence of an honest majority with less than <i>n<sup>2</sup></i> total corruptions during a refresh period; an adversary corrupting a single additional party, even if only passively, obtains the secret. This work is <i>the first feasibility result demonstrating PSS tolerating a dishonest majority</i>, it introduces the first PSS scheme secure against <i>t<n</i> passive adversaries without recovery of lost shares, it can also recover from honest faulty parties losing their shares, and when tolerating <i>e</i> faults the scheme tolerates <i>t<n-e</i> passive corruptions. A non-robust version of the scheme can tolerate <i>t<n/2-e</i> active adversaries, and mixed adversaries that control a combination of passively and actively corrupted parties that are a majority, but where less than <i>n/2-e</i> of such corruptions are active. We achieve these high thresholds with <i>O(n<sup>4</sup>)</i> communication when sharing a single secret, and <i>O(n<sup>3</sup>)</i> communication when sharing multiple secrets in batches.