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ABC has a fairly detailed analysis of mail-in ballot rejection rated in the 2016 and 2018 elections. A couple of Texas counties stand out on the map they show as having 100% rejection rate, namely Val Verde county and Hardeman county, albeit the latter with a small raw count of only 111 such votes (the former shown below with 839):

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Is anything more known why these Texas counties had such a high rejection rate for mail-in ballots in 2016? It seems peculiar because in 2018 the raw number of mail-in ballots was almost the same in Val Verde county (842), but the rejection rate was only 3.2%. In Hardeman county the rejection rate dropped to 0% in 2018 with 58 mail-in ballots cast. (I'm sparing you the map screenshot[s] for 2018 here.) ABC notes that the national trend was actually in the opposite direction, with more mail-in ballots rejected in 2018 [425,000 out of 30 million] compared to 2016 [300,000 out of 33 million]. (There is a third county in Texas with 100% rejection rate in 2018, but there were only 4 mail-in votes involved, so that's not a terribly noteworthy event.)

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This looks to be a data entry error either by the Election Administration and Voting Survey administrators, or by the county-level respondents to the EAVS, followed by ABC reporting the erroneous data without error checking.

The issue can be seen if we download the data files from which ABC's graphic draws - these can be found here. If we look at the two counties you identified which show a 100% rejection, Val Verde and Hardeman, we can then look at their responses to questions C4a and C4b, which should be the number of civilian absentee ballots that were counted and rejected, respectively, in the November 2016 general election.

For both Val Verde and Hardeman counties, these boxes both read 839 or 111 respectively, which is clearly an error, as the two boxes should sum to the response to question C1b; the number of civilian absentee ballots that were returned by voters and submitted for counting.

Thankfully, as later questions in the survey (section C5) ask for the number of ballots that were rejected for a list of reasons, we can calculate the real number of ballots that were rejected by summing these columns: 17 ballots were rejected of 839 submitted for counting in Val Verde county, while 2 were rejected of 111 submitted for counting in Hardeman county, giving us actual rejection rates of around 2% for both counties - much more reasonable!

Below is an extract from the data set showing the mentioned columns, with the two counties, as well as an example of a Texan county without such a data error - tl;dr: the C4b column has the wrong numbers in for these two counties; it should be equal to the sum of the C5 columns, and C4a+C4b should equal C1b.

| State | JurisdictionName | C1a (ballots sent out) | C1b (ballots submitted for counting) | C4a (ballots counted) | C4b (ballots rejected) | Sum of C5 columns |
|-------|------------------|------------------------|--------------------------------------|-----------------------|------------------------|-------------------|
| TX    | ARANSAS COUNTY   | 915                    | 897                                  | 894                   | 3                      | 3                 |
| TX    | HARDEMAN COUNTY  | 120                    | 111                                  | 111                   | 111                    | 2                 |
| TX    | VAL VERDE COUNTY | 1025                   | 839                                  | 839                   | 839                    | 17                |
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