Typically, high correlations between items supposed to measure the same construct would be interpreted as a sign of high reliability (small amounts of measurement error). According to Campbell and Fiske (1959), a requirement for convergent validity is that independent methods of measurement for the same construct (e.g., self and other ratings) be substantially correlated. Of course, an argument could be made to say different items constitute "different methods of measurement." So the difference between "internal consistency" and "convergent validity" is not always 100% clear.

Think of it that way: Some of the lack of "internal consistency" could stem from systematic method (indicator-specific) variance rather than random measurement error (unreliability). In other words, it is often difficult to separate invalidity from unreliability unless you have a multimethod measurement design where each construct is measured with both multiple items and multiple methods (e.g., raters) as suggested by Campbell and Fiske (1959).

Also, a scale can have high internal consistency (highly correlated items) simply due to a high degree of redundancy of the items (e.g., when asking people virtually the same questions multiple times). Nonetheless, the scale may show low or zero convergent validity relative to another assessment method. For example, a self-report measure of intelligence asking individuals to rate themselves on items such as "I am smart", "I am intelligent", etc. may show high internal consistency but low convergent validity relative to an objective intelligence test.

Campbell, D. T., & Fiske, D. W. (1959). Convergent and discriminant validation by the multitrait-multimethod matrix. Psychological Bulletin, 56(2), 81-105.